TITLE MARCH 2012 - Pakistan Academy of Sciences

Vol. 49(1), March 2012

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Proceedings of the Pakistan Academy of Sciences 49 (1): 1-8 (2012) 

Copyright © Pakistan Academy of Sciences 

ISSN: 0377 - 2969 

Pakistan Academy of Sciences 

Original Article 

On Regions of Variability of Some Differential Operators 

Implying Starlikeness 

Sukhwinder Singh Billing* 

Department of Applied Sciences 

Baba Banda Singh Bahadur Engineering College 

Fatehgarh Sahib-140 407, Punjab, India 

Abstract: In this paper, we prove a subordination theorem and use it to extend the regions of variability of 

some differential operators implying starlikeness of normalized analytic functions. Mathematica 7.0 is 

used to show the extended regions of the complex plane. 

Keywords: Analytic functions, Starlike functions, Differential subordination. 

2000 Mathematical Subject Classification: Primary 30C80, Secondary 30C45. 

1. INTRODUCTION AND 

PRELIMINARIES 

Let be the class of functions f , analytic in the 

open unit disk E { z:| z| 1} and normalized by 

the conditions f(0) f(0) 1 0 . Denote by 

* ( ), the class of starlike functions of order 

which is analytically defined as follows: 

 

* zf () 

z 

 

( ) f 

: 

, zE, 0 1 . 

f() 

z 

 

* 

We write 

* (0) , the class of univalent 

starlike functions w.r.t. the origin. Obtaining 

different criteria for starlikeness of an analytic 

function has always been a subject of interest. A 

number of criteria for starlikeness of analytic 

functions have been developed. We state below 

some of them. 

Miller et al [4] studied the class of -convex 

functions and proved the following result. 

Theorem 1.1. If a function f satisfies the 

differential inequality 

zf ( z) zf ( z) 

 

(1 ) 

1 0, z E, 

f ( z) f ( z) 

 

where is any real number, then f is starlike in 

E. 

Later on, Fukui [1] proved the more general 

result given below for the class of -convex 

functions. 

Theorem 1.2. Let , 0 be a given real 

number. For all z E , let a function f 

satisfy 

zf ( z) zf ( z) 

 

(1 ) 1 

f ( z) f ( z) 

 

 

,0 1/ 2, 

2(1 ) 

 

(1 ) ,1/ 2 1. 

2 

Then 

f * ( ) 

. 

Lewandowski et al [2] proved the following 

result. 

_____________________ 

Received, February 2011; Accepted, March 2012 

*Email: ssbilling@gmail.com

2 Variability of Some Differential Operators Implying Starlikeness 

Theorem 1.3. For a function f , the 

differential inequality 

zf ( z) zf ( z) 

 

1 0, z E, 

f ( z) f ( z) 

 

ensures the membership for f in the class 

In 2002, Li and Owa [3] proved the following 

two results: 

Theorem 1.4. If f satisfies 

zf ( z) zf ( z) 

 

 

 

1 

, z E, 

f ( z) f ( z) 

2 

for some , 0 , then 

* 

f . 


2 

zf ( z) zf ( z) 

(1 ) 

 

1 

, z E, 

f ( z) f ( z) 

4 

for some , 0 

2 , then 

f * ( / 2) . 

Later on Ravichandran et al. [10] proved the 

following result: 


zf ( z) zf ( z) 

 

1 

 

f ( z) f ( z) 

 

1 

, z E, 

2 

2 

for some , , 0 , 1, then 

* . 

f * ( ) 

. 

For more such results, we refer the readers to 

[5, 7, 9]. Recently, Singh et al [11] proved the 

following more general result for starlikeness 

which unifies all the above mentioned results. 

Theorem 1.7. Let , 0, ,0 1, and 

,0 

1, be given real numbers. Let 

M ( , , ) [1 (1 )] 

 

(1 )(1 ) (1 ) 

2 2 

2 

(1 ) , 

and 

N( , , ) [1 (1 )] 

2 (1 )(1 ) 

(1 ) 

2 

 

[2 (1 2 )(1 )(3 2 ) 

 

2(1 ) 

2 

 

(1 )(3 2 )]. 

(i) For 0 

1/ 2 , let a function f A, 

f( z) 

0 in E, satisfy 

z 

(a) 

zf ( z) zf ( z) 

 

1 

 

 

f ( z) f ( z) 

 

 

 

 

M 

( , , ), 

zf ( z) zf ( z) 

 

 

1 

1 

 

 

f ( z) f ( z) 

 

whenever 

3 4 3 

(2 13 2 ) (3 2 ) 0, and 

(b) 

zf ( z) zf ( z) 

 

1 

 

 

f ( z) f ( z) 

 

 

 

 

N( , , ), 

zf ( z) zf ( z) 

 

 

1 

1 

 

 

f ( z) f ( z) 

 

whenever 

 

3 4 3 

(2 13 2 ) (3 2 ) 0. 

Then 

f * ( ) 

. 

(ii) For 1/ 2 

1, if a function f A, 

f( z) 

0 in E, satisfies 

z 

zf ( z) zf ( z) 

 

1 

 

 

f ( z) f ( z) 

 

 

 

 

M 

( , , ), 

zf ( z) zf ( z) 

 

 

1 

1 

 

 

f ( z) f ( z) 

 

then 

f * ( ) 

. 

The main objective of this paper is to extend 

the region of variability of above mentioned 

differential operators implying starlikeness. The 

extended regions are shown pictorially using 

Mathematica 7.0.

Sukhwinder Singh Billing 3 

To prove our main results, we use the 

technique of differential subordination and need 

the following lemma of Miller and Mocanu [6]. 

For two analytic functions f and g in the 

unit disk E, we say that a function f is 

subordinate to a function g in E and write f g 

if there exists a Schwarz function w analytic in E 

with w(0) 0 and | w( z) | 1, z E such that 

f ( z) g( w( z)), z E. In case the function g is 

univalent, the above subordination is equivalent to 

f(0) g(0) 

and f(E) g(E). 

Let :CC C be an analytic function, 

p be an analytic function in E, with 

p( z), zp( z) 

C C for all z E and let h be 

 

 

univalent in E, then the function p is said to 

satisfy first order differential subordination if 

( p( z), zp( z)) h( z), ( p(0),0) h(0). 

(1) 

A univalent function q is called a dominant of 

the differential subordination (1) if p(0) q(0) 

and p q for all p satisfying (1). A dominant q 

that satisfies q q for each dominant q of (1), is 

said to be the best dominant of (1). 

Lemma 1.1. ([6], p.132, Theorem 3.4 h) Let q be 

univalent in E and let and be analytic in a 

domain D containing q ( E) 

, with ( w) 0 , 

|when w q( E) 

. Set Q( z) zq( z) [ q( z)] 

, 

h( z) [ q( z)] Q( z) 

and suppose that either 

(i) h is convex, or 

(ii) Q is starlike. 

In addition, assume that 

zh() 

z 

(iii) 0, z E . 

Qz () 

If p is analytic in E, with 

p(0) q(0), p(E) D and 

[ p( z)] zp( z) [ p( z)] [ q( z)] zq( z) [ q( z)], 

then p( z) q( z ) and q is the best dominant. 

2. MAIN RESULTS 

Theorem 2.1. Let a , b , c and d be complex 

numbers such that c and d are not simultaneously 

zero. Let qq , ( E) D, 

be a univalent function in 

E such that 

zq ( z) zq( z) czq( z) 

 

( i) 1 

0, 

q( z) q( z) cq( z) 

d 

and 

zq( z) zq( z) czq( z) 

 

1 

q( z) q( z) cq( z) 

d 

 

( ii) 

0. 

( a 2 bq( z)) q( z) 

 

 

cq( z) 

d 

 

 

 

If p, p( z) 0, z E , satisfies the differential 

subordination 

d 

ap z b p z c zp 

z 

pz ( ) 

2 

( ) ( ( )) ( ) 

d 

aq z b q z c zq 

z 

qz ( ) 

2 

( ) ( ( )) ( ), 

then p( z) q( z ) and q is the best dominant. 

(2) 

Proof. Let us define the functions and as 

follows: 

2 

d 

( w) aw bw , and ( w) c 

. 

w 

Obviously, the functions and are analytic 

d 

in domain D C 

 

c and ( w) 

0 in D. 

Now, define the functions Q and h as 

follows: 

d 

Q( z) zq( z) ( q( z)) c zq( z), 

and 

qz () 

d 

h z aq z b q z c zq 

z 

qz () 

2 

( ) ( ) ( ( )) ( ). 

Now Q is starlike in E in view of condition 

(i) and the condition (ii) implies that 

z h() 

z 

0, z E. Also by (2), we have 

Qz ()


[ p( z)] zp( z) [ p( z)] 

[ q( z)] zq( z) [ q( z)]. 

Therefore, the proof, now, follows from 

Lemma 1.1. 

zf () 

z 

Setting pz () in Theorem 2.1, we 

f() 

z 

have the following result. 

Theorem 2.2. Let q, q( z) 0, be a univalent 

function in E and satisfy the conditions (i) and (ii) 

zf () 

z 

of Theorem 2.1. If f , 0, z E , 

f() 

z 

satisfies the differential subordination 

zf ( z) f ( z) 

 

a b c d 

zf ( z) 

f ( z) zf ( z) 

 

f( z) 

zf ( z) zf ( z) 

 

1 

 

 

f ( z) f ( z) 

 

2 d 

aq( z) b( q( z)) c zq( z), 

qz () 

where a , b , c and d are complex numbers, 

zf () 

z 

then qz () and q is the best dominant. 

f() 

z 

If we restrict the constants a , b , c and d to 

real numbers. By selecting a 1 (1 ), 

b(1 ), c(1 ) and d in 

Theorem 2.2, we obtain the following result. 

Theorem 2.3. Let q, q( z) 0, be a univalent 

function in E and satisfy the conditions 

(i) 

(ii) 

zq( z) zq( z) 

 

1 

q( z) q( z) 

 

 

0, 

and 

(1 ) zq( z) 

 

 

(1 ) qz ( ) 

 

 

zq ( z) zq( z) (1 ) zq( z) 

1 

q( z) q( z) (1 ) q( z) 

 

[(1 (1 )) 2 (1 ) q( z)] q( z) 

 

 

0. 

(1 ) qz ( ) 

 

zf () 

z 

If f , 0, z E, satisfies the 

f() 

z 

differential subordination 

zf ( z ) ( ) ( ) ( ) 

1 

zf z 

1 zf z 

1 

zf z 

 

f ( z) f ( z) f ( z) f ( z) 

 

[1 (1 )] ( ) (1 )( ( )) 

2 

q z q z 

 

1 

zq( z), 

qz () 

where and are real numbers, then 

zf () 

z 

qz () and q is the best dominant. 

f() 

z 

3. APPLICATIONS TO STARLIKE 

FUNCTIONS 

Throughout this section, we restrict the constants 

a , b , c and d to real numbers. 

Remark 3.1. When we select 

1 (1 2 ) 

z 

qz ( ) , 0 1. Then 

1 

z 

zq ( z) zq( z) z (1 2 ) 

z 

1 1 . 

q( z) q( z) 1 z 1 (1 2 ) 

z 

Thus, 

zq 

( z) zq( z) 

 

1 

0. 

q( z) q( z) 

 

Also 

zq( z) zq( z) 1 z 

1 qz ( ) 1 

q( z) q( z) 1 

z 

(1 2 ) z 1 1 (1 2 ) z 

 

. 

1 (1 2 ) z 1 

z 

For 0, we have 

zq 

( z) zq( z) 1 

1 qz ( ) 

0. 

q( z) q( z) 

 

Therefore, qz () satisfies the conditions of 

Theorem 2.2 in case where a 1, b 0, c 0 and 

d 

, 0 and we get the following result.


Corollary 3.1. Let be a real number with 

zf () 

z 

0 . If f , 0, zE 

, satisfies the 

f() 

z 

differential subordination 

zf ( z) zf ( z) 

 

(1 ) 1 

 

f ( z) f ( z) 

 

1 (1 2 ) z 2 (1 ) 

z 

 

, 

1 z (1 z)(1 (1 2 ) z) 

can vary over the portion of the plane right to the 

curve h () 1 

z for the same conclusion. Thus our 

result extends the region of variability of this 

operator for the same implication and the region 

bounded by the dashing line and the curve is the 

claimed extension as shown in Fig. 1. 

then 

f 

* ( ) 

. 

1 3 

Remark 3.2. For and , Corollary 

2 4 

3.1 reduces to the following result. 

zf () 

z 

If f , 0, zE 

, satisfies the 

f() 

z 

condition 

zf ( z) zf ( z) 2 z z 

1 h1 

( z), 

f ( z) f ( z) 1 z (1 z)(2 z) 

then 

f * (3/ 4) . 

Fig. 1. 

Substituting the same values of and in 

the result of Fukui [1] stated in Theorem 1.2, we 

obtain the following result: 

If f 

, satisfies the condition 

zf ( z) zf ( z) 4 

1 , z E, 

f ( z) f ( z) 3 

then 

f * (3/ 4) . 

To compare both the results, we plot h ( ) 1 

4 

and the line ( z) 

in Fig. 1. 

3 

We see that according to the result of Fukui 

[1], for the starlikeness of order 3/ 4 of f() 

z , 

zf ( z) zf ( z) 

the differential operator 1 

can 

f ( z) f ( z) 

vary in the complex plane on the right side of the 

4 

line ( z) 

shown with dashes in Fig. 1 

3 

whereas according to our result, the same operator 

Remark 3.3. When we select 

1 (1 2 ) 

z 

qz ( ) , 0 1. Then 

1 

z 

zq 

( z) 1 z zq 

( z) 

 

1 , i.e. 1 0, 

q 

 

( z) 1 z q( z) 

 

and 

zq( z) 1 

1z 

1 2 qz ( ) 

q( z) 1 

z 

1 (1 2 ) z 1 

2 . 

1 

z 

Therefore for 0 

1, we have 

zq( z) 1 

 

1 2 qz ( ) 

0. 

q() 

z



Theorem 2.2 for a 1 , b , 

c and 

d 0 and we obtain the following result. 

Corollary 3.2. Let be a real number 

zf () 

z 

0 

1. If f , 0, 

z E , satisfies 

f( 

z) 

the differential subordination 

zf ( z) zf ( z) 

 

1 

 

f ( z) f ( z) 

 

1 (1 2 ) z 1 (1 2 ) 

z 

1 

 

1z 

 

1z 

2 (1 ) 

z 

 

, 

(1 z)[1 (1 2 ) z] 

 

then 

f * (1/ 2) . 

then 

f 

* ( ) 

. 

Fig 2. 

Note that for 1 and 0 in above 

corollary, we get Theorem 1 of Nunokawa et al 

[8]. 

Remark 3.4. For 1 and 

reduces to the following result. 

1 

, Corollary 3.2 

2 

zf () 

z 

f , 0, 

z E , satisfying the condition 

f( 

z) 

zf ( z) zf ( z) 

 

1 

 

f ( z) f ( z) 

 

1 

z 

h z f 

2 

(1 z) 

* 

2( ), (1/ 2). 

Substituting the same values of and in 

the result of Ravichandran [10] stated in Theorem 

1.6, we obtain the following result. 

If f 


zf ( z) zf ( z) 

 

1 0, z E, 

f ( z) f ( 

z) 

 

To compare both the results, we plot h ( E) 

in 2 

Fig. 2 and we see that according to the result of 

Ravichandran, for the starlikeness of order 1/ 2 of 

f() 

z , the differential operator 

zf ( z ) ( ) 

1 

zf z 

can vary in the right half 

f ( z) f ( z) 

 

complex plane whereas according to our result, the 

same operator can vary over the portion of the 

plane bounded by the curve h () 2 

z (entire shaded 

regiom) for the same conclusion. Thus shaded 

portion in the left half plane as shown in Fig. 2, is 

the extension of the region of variability of this 

operator for the same implication. 

1 

Remark 3.5. When we select qz () 1 z 

. Then 

zq( z) zq( z) zq( z) 

 

1 

q( z) q( z) q( z) 1 

2 

2 z 

 

0, 

(1 z)(2 z) 

 

and


zq( z) zq( z) zq( z) (1 2 q( z)) q( z) 

 

1 

q( z) q( z) q( z) 1 q( z) 1 

 

2 

5 zz 

 

 

0. 

(1 z)(2 z) 

 

region of variability of this operator for the same 

implication. 


1 

Theorem 2.3 for 1 and and we obtain 

2 

the following result: 

zf () 

z 

Corollary 3.3. If f , 0, z E , 

f() 

z 

satisfies the differential subordination 

2 

zf ( z) zf ( z) 2 z z 

1 1 h 

2 3( z), 

f ( z) f ( z) (1 z) 

zf ( z) 1 

then 

, z E, i.e. 

f ( z) 1 

z 

f * (1/ 2) . 

1 

Remark 3.6. When we replace 1 and , 

2 

Theorem 1.7 of Singh et al [11], we obtain the 

following result. 

If f 


zf ( z) zf ( z) 

 

1 1 0, z E, 

f ( z) f ( 

z) 

 

then 

f * (1/ 2) . 

To compare this result with Corollary 3.3, we 

plot h (E) 3 

in Fig. 3 and we see that according to 

the result of Singh et al [11], for the starlikeness of 

order 1/ 2 of f() 

z , the differential operator 

zf ( z) zf ( z) 

 

1 

1 

can vary in the right 

f ( z) f ( z) 

 

half complex plane whereas according to the result 

in Corollary 3.3, the same operator can vary over 

the portion of the plane bounded by the curve 

h () 3 

z (whole shaded region) for the same 

conclusion. Thus shaded portion in the left half 

plane as shown in Fig. 3, is the extension of the 

Fig. 3. 

4. REFERENCES 

1. Fukui, S. On -convex functions of order β. 

Internat. J. Math. & Math. Sci. 20 (4): 769–772 

(1997). 

2. Lewandowski, Z., S.S. Miller & E. Zlotkiewicz. 

Generating functions for some classes of univalent 

functions. Proc. Amer. Math. Soc. 56: 111–117 

(1976). 

3. Li, J.-L. & S. Owa. Sufficient conditions for 

starlikeness, Indian J. Pure Appl. Math. 33: 313– 

318 (2002). 

4. Miller, S.S., P.T. Mocanu & M.O. Reade. All - 

convex functions are univalent and starlike. Proc. 

Amer. Math. Soc. 37: 553–554 (1973). 

5. Miller, S.S., P.T. Mocanu, & M.O. Reade. 

Bazilevic functions and generalized convexity. 

Rev. Roumaine Math. Pures Appl. 19: 213–224 

(1974). 

6. Miller, S.S. & P.T. Mocanu. Differential 

Suordinations: Theory and Applications. Series on 

Monographs and Textbooks in Pure and Applied 

Mathematics (No. 225). Marcel Dekker, New York 

(2000). 

7. Mocanu, P.T. Alpha-convex integral operators and 

strongly starlike functions. Studia Univ. Babes- 

Bolyai Math. 34 (2): 18–24 (1989). 

8. Nunokawa, M., N.E. Cho, O.S. Kwon, S. Owa, & 

S. Saitoh. Differential inequalities for certain 

analytic functions. Compt. Math. Appl. 56: 2908– 

2914 (2008).


9. Padmanabhan, K.S. On sufficient conditions for 

starlikeness. Indian J. Pure Appl. Math. 32 (4): 

543–550 (2001). 

10. Ravichandran, V., C. Selvaraj & R. Rajalakshmi. 

Sufficient conditions for starlike functions of order 

. J. Inequal. Pure and Appl. Math. 3 (5): Art. 81: 

1–6 (2002). 

11. Singh, S., S. Gupta, & S. Singh. Starlikeness of 

analytic maps satisfying a differential inequality. 

General Mathematics 18 (3): 51–58 (2010).



ISSN: 0377 - 2969 



Some New s-Hermite-Hadamard Type Inequalities for 

Differentiable Functions and Their Applications 

Muhammad Muddassar*, Muhammad I. Bhatti and Muhammad Iqbal 

Department of Mathematics, University of Engineering & Technology, 

Lahore, Pakistan 

Abstract: In this paper, we establish several inequalities for some differentiable mappings that are 

connected with the celebrate Hermit - Hadamard integral inequality for s-convex functions. Also a parallel 

development is made base on concavity. Applications to some special means of real numbers are found. 

Also applications to numerical integration are provided. 

Keywords and Phrases: Hermite-Hadamard type inequality, s-Convex function, p-logarithmic mean, 

H lder’s inequality, Trapezoidal formula, special means. 

(AMS SUBJECT CLASSFICATION: 26D15, 39A10 and 26A51) 

1. INTRODUCTION 

Let a function defined as 

is 

said to be convex if the following inequality holds 

For all x, y and t [0, 1]. Geometrically, 

this means that if P, Q and R are three distinct 

points on graph of f(x) with Q between P and R, 

Then Q is on or below chord PR. There are many 

result associated with convex function in the area 

of inequalities, but one of those is the classical 

Hermite Hadamard inequality. 

for . 

(1.1) 

Hudzik and Maligranda [3] considered, among 

others, the class of functions which are s-convex 

in the second sense. This is defining as follows. 

A function is said to be s- 

convex in the second sense if 

_____________________ 

(1.2) 

Received, December 2010; Accepted, March 2012 

*Corresponding author: Muhammad Muddassar; Email: malik.muddassar@gmail.com 

holds for all 

and for some 

fixed s that every 1- 

convex function is convex. In the same Paper [3] 

H. Hudzik and L. Maligranda discussed a few 

result connecting with s-convex function in second 

sense and some new result about Hadamard 

inequality for s-convex function is discussed in [2, 

7]. On the hand, there are many important 

inequalities connecting with 1-convex (Convex) 

function [2], but one of these is (1.1). 

Dragomir et al [7], proved a variant of Hermit- 

Hadamard inequality for s-convex function in 

second sense. 

Theorem 1. Suppose that is s- 

convex function in the second sense. Where 

and let 

then the following inequality holds. 

(1.3) 

The constant is the best possible in 

the second inequality in (1.3). The inequality in 

(1.3) becomes reverse when the function is

10 Muhammad Muddassar et al 

concave. The result in (1.3) was improved by 

Jagers [4] who gave both the upper and lower 

bounds for the constant c(s) in the inequality 


Theorem 2. Let 

be differentiable 

function on , with If 

if the mapping is s-convex on 

[a, b], then 

He proved that 

Dragomir et al [2] discussed inequality for 

differentiable and twice differentiable function 

connecting with the Hermite – Hadamard (H-H) 

Inequality in the basis of the following Lemmas. 

Proof. From Lemma 1, 

(2.6) 

Lemma 1. Let 


function on (interior of , 

(1.4) 

Dragomir and Agarwal [1] established the 

following result connected with the right part of 

(1.4) as well as to apply them for some elementary 

inequalities for real numbers and numerical 

integration. 

Where 

is s-convex on [a, b] for t 

(2.7) 

Lemma 2. Let 


function on , with If 

By (2.8) and (2.7), we get (2.6). 

(2.8) 

(1.5) 

This paper is organized as follows: after 

Introduction, we discuss some new s-Hermite 

Hadamard type inequalities for differentiable 

function in section 2, and in section 3 we 

give some applications of the results from section 

2 for some special means of real numbers. In 

section 4, we give some application, to trapezoidal 

formula. 

Theorem 3. Let the assumptions of Theorem 2 are 

satisfied with p > 1 such that . If the 

mapping 

is concave on [a, b] then, 

. (2.9) 

Proof. From Lemma 1,

Inequalities for Differentiable Functions 11 

(2.10) 

And 

By applying H lder’s inequality on right side of 

(2.10). We have; 

(2.17) 

Here 

(2.11) 

(2.12) 

By (2.16) and (2.17), we get (2.14). 

Corollary 5. From theorem 4, the assumptions of 

theorem 2 are satisfied with p > 1 such that 

. If the mapping is -convex 

on 

then 

Since is concave, by applying Jensen’s 

Integral Inequality on the second integral of 

R.H.S. of (2.11). We have 

By (2.10), (2.12) and (2.13).We get (2.14). 

(2.13) 

Theorem 4. Let the assumptions of theorem 2 are 


mapping is -convex on then 

Proof. The above inequality is obtained by using 

the fact 

for ) with 



mapping is s-concave on then 


(2.14) 

(2.15) 

(2.18) 

Proof. We proceed similarly as in theorem 4. 

By of we obtain 

. (2.19) 

Now (2.18) immediately follows from theorem 1. 

By applying H lder’s inequality on right side of 

(2.15). We get 


satisfied, we have another result: 

(2.16) 

Since is s-convex on [a, b] for t , then 

(2.20)



Proof. The proof is similar to that of corollary 5. 

(2.21) 

By applying H lder’s inequality on (2.21) for q > 

1, we have 



mapping 

is s-concave on [a, b], then 

(2.22) 

By s-convexity of on [a, b] for all . 

(2.22) can be written as: 

Proof. We proceed similarly as in theorem 6. 

By of we obtain 

(2.25) 

(2.26) 

Now (2.25) immediately follows from theorem 1. 

Here, 

(2.23) 

Theorem 10. Let the assumptions of theorem 2 

are satisfied, then 

(2.24) 

By (2.23) and (2.24) in (2.21), we get (2.20). 

Proof. From Lemma 2. 

(2.27) 

Corollary 8. From theorem 7, the assumptions of 

theorem 4 are satisfied with p > 1 such that 

. If the mapping is -convex on 

then


(2.32) 

(2.28) 

By applying H lder Inequality in (2.32), we have 

By using s-convexity of on [a, b] for all 

on right side of (2.28), we have 

(2.33) 

(2.29) 

But 

But 

(2.30) 

By (2.29) and (2.30) we get (2.27). 

Theorem 11. Let the assumptions of Theorem 2 

are satisfied. Furthermore, if the mapping is 

concave on [a, b] for q > 1, then 

(2.34) 

Since is concave on [a, b] so by using 

Jensen’s Integral Inequality on first integral in 

R.H.S., we have 

Proof. From Lemma 2, we have 

(2.31) 

= (2.35) 

Hence (2.33), (2.34) and (2.35) together imply 

(2.31). 



s-convex on [a, b] for then


And 

(2.41) 


By (2.39), (2.40) and (2.41), we have (2.36). 

Corollary 13. From theorem 12, Let the 

assumptions of Theorem 2 are satisfied. 

Furthermore, if the mapping is s-convex on 

[a, b] for then 


By applying H lder Inequality, (2.37) becomes 

(2.37) 



s-concave on for then 

By s-convexity of on for 

we have 

(2.38) 

(2.42) 

Proof. We proceed in a similar way as in theorem 

10. 

By s-concavity of | f’ | q we obtain 

(2.43) 

Now (2.42) immediately follows from Theorem 1. 

But 

(2.39) 



function of , a, b, with a < b, and 

if the mapping is s-convex on 

then 

(2.40)


By solving (2.48), we have 


(2.44) 

(2.49) 

Relations (2.46), (2.47), and (2.49) together imply 

(2.44). 

Corollary 16. From theorem 15, Let 

be differentiable function of , a, b, with a 

< b, and if the mapping is s- 

convex on for then 

(2.45) 

By applying H lder inequality on (2.45), we 

follow as 




function on , a , b with a < b, and 

If the mapping is s-concave 

on for then 

Here 

And 

(2.46) 

(2.47) 

(2.50) 

Proof. We proceed in a similar way as in theorem 

12. 

By 

, we obtain 

Since 

(2.48) 

(2.51) 

Now (2.50) immediately follows from theorem 1.


3. APPLICATION TO SOME SPECIAL 

MEANS 

Let us recall the following means for any two 

positive numbers a and b. 

(1) The Arithmetic mean 

(2) The Harmonic mean 

(3) The p- Logarithmic mean 

Proposition 2. Let p > 1, 0 < a 

then 

Proof. Following by Theorem 12, setting 

for 

Another result which is connected with p- 

Logarithmic mean is the following one. 

Proposition 3. Let p > 1, 0 < a 

then 

(4). The Identric mean 

Proof. Following by Theorem 15, setting 

and for 

(5). The Logarithmic mean 

The following inequality is well known in the 

literature in [3]: 

It is also known that 

over 

. 

monotonically increasing 

Now here we find some new applications for 

special means of real numbers by using the results 

of Section 2. 

Proposition 1. Let p > 1, 0 < a 

Then one has the inequality. 

(3.52) 

Proof. By theorem 10 applied for the mapping 

for we have the above inequality 

(3.52). 

4. APPLICATION TO QUADRATURE 

FORMULAE 

Let be a division 

of the interval [a, b] and 

consider the quadrature formula 

where, for the trapezoidal version 

and the connected error term 

trapezoidal version 

is 

(4.53) 

for the 

Proposition 4. Let be 

differentiable function on such that 

, where with s- 

convex on [a, b], for every division D of [a, b], the 

trapezoidal error estimate satisfies 

(4.54)


Where 

Proof. On applying Corollary 8 on the subinterval 

[ ] of the division D of [a, b] for 

, we have 

(4.55) 

Taking sum over from . And using 

s- convexity of , we get, 

Using (4.55) and (4.56), we get (4.54). 

(4.56) 

Proposition 5. Let 


function on such that , where 

with s- convex on [a, b] 

, for every division D of [a, b], the trapezoidal 

error estimate satisfies 

Where 

Proof. The proof is similar to that of Proposition 4 

and using Corollary 16. 

5. CONCLUSIONS 

By selecting some other convex function, and 

applying the results given in section 2, we can find 

out some new relations connecting to some special 

means. For example, choosing different convex 

function like and 

for different values of s 

from (0, 1] in s-convexity (concavity), we get new 

relation relating to some special means. 

6. REFERENCES 

1. Dragomir, S.S. & R.P. Agarwal. Two inequalities 

for differentiable mappings and applications to 

special means of real numbers and trapezoidal 

formula. Applied Mathematics Letter 11 (5): 91–95 

(1998). 

2. Dragomir, S.S. & C.E.M. Pierce. Selected Topics 

on Hermite-Hadamard Inequalities and 

Applications. RGMIA, Monographs, Victoria 

University. (online: http://ajmaa.org/ RGMIA/ 

monographs.php/) (2000). 

3. Hudzik, H. & L. Maligranda. Some remarks on s- 

convex functions. Aequationes Mathematicae 48: 

100–111 (1994). 

4. Jagers, B. On a hadamard-type inequality for s- 

convex functions. http://wwwhome.cs.utwente.nl/ 

jagersaa/alphaframes/Alpha.pdf. 

5. Kavurmaci, H., M. Avci & M.E. Özdemir. New 

inequalities of Hermite-Hadamard type for convex 

functions with applications. Journal of Inequalities 

and Applications, Art No. 86 doi:10.1186/ 1029- 

242X-2011-86 (2011). 

6. Kurmaci, U.S. Inequalities for differentiable 

mappings and applications to special means of real 

numbers and to midpoint formula. Applied 

Mathematics Computation 147 (1): 137–146 

(2004). 

7. Kurmaci , U.S. & M.E. Özdemir. On some 

inequalities for differentiable mappings and 

applications to special means of real numbers and 

to midpoint formula. Applied Mathematics 

Computation 153 (2): 361–368 (2004). 

8. Avci, M., H. Kavurmaci & M.E. Özdemir. New 

inequalities of Hermite–Hadamard type via s- 

convex functions in the second sense with 

applications. Applied Mathematics Computation 

217 (12): 5171–5176 (2011). 

9. Pearce, C.E.M. & J. Pěcarić. Inequalities for 

differentiable mappings with application to special 

means and quadrature formulae. Applied 

Mathematics Letter 13 (2): 51–55 (2000). 

10. Pěcarić, J., F. Proschan & Y.L. Tong. Convex 

Functions, Partial Ordering and Statistical 

Applications. Academic Press, New York (1991).

18 Muhammad Muddassar et al



ISSN: 0377 - 2969 



Supra β-connectedness on Topological Spaces 

O.R. Sayed* 

Department of Mathematics, Faculty of Science, 

Assiut University, Assiut 71516, Egypt 

Abstract. In this paper, supra β-connectedness are researched by means of a supra β- separated sets. 

Keywords and Phrases: Supra β -separated, supra β -connectedness and supra topological space. 

2000 Mathematics Subject Classification: 54D05. 


Some types of sets play an important role in the 

study of various properties in topological spaces. 

Many authors introduced and studied various 

generalized properties and conditions containing 

some forms of sets in topological spaces. In 1983, 

Mashhour et al [2] developed the supra topological 

spaces and studied S-continuous maps and 

S * −continuous maps. We will use the term supracontinuous 

maps instead of S-continuous maps. In 

2008, Devi et al [1] introduced and studied a class 

of sets and maps between topological spaces 

called supra α−open sets and supra α−continuous 

maps, respectively. In 2010, Sayed and Noiri [4] 

introduced the concepts of supra b-open sets, supra 

b-continuity, supra b-open maps and supra b- 

closed maps and studied some of their properties. 

In [3] the concepts of supra β-open sets, supra β- 

continuity, supra β-open maps and supra β-closed 

maps were introduced and some of their properties 

were investigated. The purpose of this paper is to 

introduce the concept of supra β-connectedness 

based on supra β-separated sets. We prove that 

supra β-connectedness is preserved by supra β- 

continuous bijections. 

Throughout this paper, (X, τ ) , (Y, σ) and (Z, 

υ) (or simply, X , Y and Z) denote topological 

spaces on which no separation axioms are 

assumed unless explicitly stated. All sets are 

assumed to be subsets of topological spaces. The 

closure and the interior of a set A are denoted by 

_____________________ 

Cl(A) and Int(A), respectively. A sub collection μ 

2 X is called a supra topology [2] on X if X μ 

and μ is closed under arbitrary union. (X, μ) is 

called supra topological space. The elements of μ 

are said to be supra open in (X, μ) and the 

complement of a supra open set is called supra 

closed. The supra closure of a set A, denoted by 

Cl μ (A), is the intersection of supra closed sets 

including A. The supra interior of a set A, denoted 

by Int μ (A), is the union of supra open sets included 

in A. The supra topology μ on X is associated 

with the topology τ if τ μ. A set A is supra β- 

open [3] if A Cl μ (Int μ (Cl μ (A))). The complement 

of a supra β-open set is called supra β-closed. Thus 

A is supra β-closed if and only if 

Intμ(Clμ(Intμ(A))) A. The supra β-closure of a set 

 

A [3], denoted by Cl (A), is the intersection of the 

supra β-closed sets including A. The supra β- 

 

interior of a set A [3],denoted by Int (A), is the 

union of the supra β-open sets included in A. Let 

(X, τ ) and (Y, σ) be two topological spaces and μ 

be an associated supra topology with τ . A map f : 

X → Y is called a supra β-continuous map [3] if 

the inverse image of each open set in Y is a supra 

β-open set in X. 

The following theorem was given by Ravi et 

al [3]: 

Theorem 1.1. Let (X, τ ) and (Y, σ) be two 

topological spaces and μ be an associated supra 

 

Received, June 2011; Accepted, March 2012 

*Email: o_r_sayed@yahoo.com

20 O.R. Sayed 

topology with τ . Let f be a map from X into Y. 

Then the following are equivalent: 

(1) f is a supra β-continuous map; 

(2) The inverse image of each closed set in Y is 

a supra β-closed set in X; 

(3) f -1 (A)) f -1 (Cl(A)) for every set A in Y; 

(4) f( A)) Cl(f(A)) for every set A in X; 

(5) f -1 (Int(B)) (f -1 (B)) for every B in Y . 

2. SUPRA -SEPARATED SETS 

In this section, we shall research supra β-separated 

sets in topological spaces. 

Definition 2.1. Let (X, τ ) be a topological space 

and A,B be two non-empty subsets of X. Then A 

and B are said to be supra β-separated if A ∩ 

B) = ϕ and A) ∩ B = ϕ. 

The following result is immediate from the 

above definition: 

Theorem 2.1. Let C and D are two non-empty 

subsets of the supra β- separated sets A and B, 

respectively. Then C and D are also supra β- 

separated in X. 

Theorem 2.2. Let A,B be two non-empty subsets of 

X such that A ∩ B = ϕ and A,B are either they 

both are supra β-open or they both are supra β- 

closed. Then A and B are supra β-separated. 

Proof. If both A and B are supra β-closed sets and 

A ∩ B = ϕ, then A and B are supra β-separated. Let 

A and B be supra β-open and A ∩ B = ϕ. Then A 

X −B. So A) X − B) = X – (B) = 

X − B. Hence 

A) ∩ B = ϕ. Similarly, A ∩ 

B) = ϕ. Thus A and B are supra β- separated. 

Theorem 2.3. Suppose that A and B are two nonempty 

subsets of X such that either they both are 

supra β-open or they both are supra β-closed. If C 

= A∩(X−B) and D = B∩(X−A),then C and D are 

supra β-separated, provided they are non-empty. 

Proof. First suppose A and B are both supra β- 

open. Now, D = B ∩ (X − A) implies D X − A. 

Then D) X − A) = X – (A) = X − 

A. Hence A ∩ D) = ϕ. Therefore C ∩ D) 

= ϕ. Similarly, C) ∩D = ϕ. Thus C and D are 

supra β- separated. 

Next, suppose that A and B are both supra β- 

closed sets. Then C = A ∩ (X − B), implies C A. 

Hence C) A) = A. Therefore C) ∩ 

D = ϕ. Similarly, C) ∩ D = ϕ. Thus C and D 

are supra β- separated. 

Theorem 2.4. Two non-empty subsets A and B of 

X are supra β- separated if and only if there exists 

two supra β-open sets U and V such that A U, B 

V, A ∩ V = ϕ, B ∩ U = ϕ. 

Proof. Suppose that A and B are two supra β- 

separated. Now, A ∩ B) = ϕ and A) ∩ B 

= ϕ. Then A X − B) = U (say); and B X − 

A) = V (say). Since both A) and B) 

are supra β-closed, then both U and V are supra β- 

open. Therefore A (A) = X − V and B 

B) = X − U. Hence A ∩ V = ϕ and B ∩ U = ϕ. 

Conversely, let U and V be supra β-open such 

that A U, B V, A ∩ V = ϕ and B ∩ U = ϕ. 

Then X − U and X − V are supra β-closed. Also, 

A ∩ V = ϕ implies A X − V. Thus A) 

X −V ) = X −V. Hence A) ∩ V = ϕ. 

Similarly, U ∩ 

supra β-separated. 

B) = ϕ. Thus A and B are 

3. SUPRA -CONNECTEDNESS 

In this section, we research supra β-connectedness 

by means of supra β-separated. 

Definition 3.1. A subset A of X is supra β- 

connected if it can't be represented as a union of 

two non-empty supra β-separated sets. When A = 

X is supra β-connected, then X is called supra β- 

connected space. 

Theorem 3.1. A non-empty subset C of X is supra 

β-connected if and only if for every pair of supra 

β-separated sets A and B in X with C AB, 

exactly one of the following possibilities holds: 

(a) C A and C ∩ B = ϕ, 

(b) C B and C ∩ A = ϕ.

Supra β-connectedness on Topological Spaces 21 

Proof. Let C be supra β-connected. Since C A 

B, then both C ∩ A = ϕ and C ∩ B = ϕcan not 

hold simultaneously. If C ∩ A ϕ and C ∩ B ϕ, 

then by Theorem 2.1 they are also supra β- 

separated and C = (C ∩ A) (C ∩ B) which goes 

against the supra β-connectedness of C. Now, if 

C ∩ A = ϕ, then C B, while C A holds if 

C ∩ B = ϕ. 

Conversely, suppose that the given condition 

holds. Assume by contrary that C is not supra β- 

connected. Then there exist two non-empty supra 

β-separated sets A and B in X such that C = A B. 

By hypothesis, either C ∩ A = ϕ or C ∩ B = ϕ. So, 

either A = ϕ or B = ϕ, none of which is true. Thus 

C is supra β-connected. 

Theorem 3.2. The following are equivalent: 

(1) A space X is not supra β-connected. 

(2) There exist two non-empty supra β-closed 

sets A and B such that A B = X and A∩B 

= ϕ. 

(3) There exist two non-empty supra β-open sets 

A and B such that A B = X and A∩B = ϕ. 

Proof. (1) (2): Suppose that X is not supra β- 

connected. Then there exist two non-empty 

subsets A and B such that Cl μ β(A) ∩ B = A ∩ 

Cl μ β(B) = ϕ and A B = X. It follows that A) 

= A) ∩ (AB) = ( A) ∩ A) ( A) ∩ B) 

= Aϕ = A. Hence A is supra β-closed set. 

Similarly, B is supra β-closed. Thus (2) is held.(2) 

(3) and (2) (1): Obvious. 

Corollary 3.1. The following are equivalent: 

(1) A space X is supra β-connected. 

(2) If A and B are supra β-open sets, A B = X 

and A ∩ B = ϕ, then ϕ {A,B}. 

(3) If A and B are supra β-closed sets, A B = 

X and A ∩ B = ϕ, then ϕ {A,B}. 

Theorem 3.3. For a subset G of X, the following 

conditions are equivalent: 

(1) G is supra β-connected. 

(2) There does not exist two supra β-closed sets 

A and B such that A∩G ϕ, B ∩G ϕ,G 

A B and A ∩ B ∩ G = ϕ. 

(3) There does not exist two supra β-closed sets 

A and B such that G A, G B, G A B 

and A ∩ B ∩ G = ϕ. 

Proof. (1) (2): Suppose that G is supra β- 

connected and there exist two supra β-closed sets 

A and B such that A ∩ G ϕ, B ∩ G ϕ, G A 

B and A ∩ B ∩ G = ϕ. Then (A ∩ G) ( B ∩ G) = 

(A B) ∩ G = G. Also, A ∩ G) ∩ ( B ∩ G) 

A) ∩ (B ∩G) = A∩B ∩G = ϕ. Similarly, 

(A∩G)∩ B ∩G) = ϕ. This shows that G is not 

supra β-connected, which is a contradiction. 

(2) (3): Suppose by opposite that there exist 

two supra β-closed sets A and B such that G A, 

G B, G A B and A ∩ B ∩ G = ϕ. Then A ∩ 

G ϕ and B ∩ G ϕ. This is a contradiction. 

(3) (1): Suppose that (3) is satisfied and G is 

not supra β-connected. Then there exist two nonempty 

supra β-separated sets C and D such that G 

= C D. Thus C) ∩ D =C ∩ D) = ϕ. 

Assume that 

A = C) and B = D). Hence G A B 

and C) ∩ D) ∩ (C D) = ( C) ∩ 

D) ∩ C) ( C) ∩ D) ∩ D) = 

( D)∩C) ( C)∩D) = ϕ ϕ = ϕ. Now we 

prove that G A and G B. In fact, if G A, 

then D) ∩ G = B ∩ G = B ∩ (G ∩ A) = ϕ, a 

contradiction. Thus G A. Analogously we have 

G B. This contradicts (3). Therefore G is supra 

β-connected. 

Corollary 3.2. A space X is supra β-connected if 

and only if there does not exist two non-empty 

supra β-closed sets A and B such that A B = X 

and A ∩ B = ϕ. 

Theorem 3.4. For a subset G of X, the following 

conditions are equivalent: 

(1) G is supra β-connected. 

(2) For any two supra β-separated sets A and B 

with G A B, we have G ∩ A = ϕ or G ∩ 

B = ϕ. 

(3) For any two supra β-separated sets A and B 

with G A B, we have G A or G B.

22 O.R. Sayed 

Proof. (1) (2): Suppose that A and B are supra 

β-separated and G A B. Then by Theorem 2.1 

we have G ∩ A and G ∩ B are also supra β- 

separated. Since G is supra β-connected and G = G 

∩ (A B) = (G ∩ A) (G ∩ B), then G ∩ A = ϕ 

or G ∩ B = ϕ. 

(2) (3): If G ∩ A = ϕ, then G = G ∩ (A B) = 

(G ∩ A) (G ∩ B) = G ∩ B. So, G B. Similarly, 

G ∩ B = ϕ implies G A. 

(3) (1): Suppose that A and B are supra β- 

separated and G = A B. Then by (3) either G A 

or G B. 

If G A, then B = B ∩ G B ∩ A B ∩ A) 

= ϕ. Similarly, if G B, then A = ϕ. So G can't be 

represented as a union of two non-empty supra β- 

separated sets. Therefore G is supra β-connected. 

Theorem 3.5. Let G be a supra β-connected 

subsets of X. If G H G), then H is also 

supra β-connected. 

Proof. Suppose that H is not supra β-connected. 

By Theorem 3.3 there exist two supra β-closed 

sets A and B such that H A, H B, H A B 

and A∩B ∩H = ϕ. Since G H, then G A B 

and A ∩B ∩G = ϕ. Now we prove that G A and 

G B. In fact, if G A, then G) A) = 

A. Therefore by hypothesis H A which is a 

contradiction. Hence,G A. Similarly, G B. This 

contradicts that G is supra β-connected. 

Theorem 3.6. Let G and H be supra β-connected. 

If G and H are not supra β-separated, then G H 

is supra β-connected. 

Proof. Suppose that G H is not supra β- 

connected. By Theorem 3.3 there exist two supra 

β-closed A and B such that G H A,G H 

B,G H A B and (G H)∩(A∩B) = ϕ. So, 

either G A or H A. Assume G A. Then G B 

because G is supra β-connected. Hence H B and 

H A. Thus A∩G A∩B∩(G∩H) = ϕ. Therefore 

H)∩G A)∩G =A ∩ G = ϕ. Similarly, 

H ∩ G) = ϕ. This shows that G and H are 

supra β-separated, a contradiction. 

Theorem 3.7. Let be a family of supra 

β-connected subsets of X. If there is j I such that 

and are not supra β-separated for each i j, 

then is supra β-connected. 

Proof. Suppose that is not supra β- 

connected. Then there exist two non-empty supra 

β-separated subsets A and B of X such that 

= A B. For each i I, is supra β-connected 

and A B. Then by Theorem 3.1 either 

A and ∩ B = ϕ, or else B and ∩A = ϕ. 

If possible, let for some r, s I with r s, A 

and B. Then , being non-empty of supra 

β-separated sets which is not the case. Thus either 

A with ∩ B = ϕ for each i I or else 

B with ∩ A = ϕ for each i I . In the first case 

B = ϕ (since B ) and in the second case A 

= ϕ. Non of which is true. Thus is supra β- 

connected. 

Corollary 3.3. Let be a family of supra 

β-connected sets. If ϕ, then is 

supra β-connected. 

Theorem 3.8. A non-empty subset G of X is supra 

β-connected if and only if for any two elements x 

and y in G there exists a supra β-connected set H 

such that x, y H G. 

Proof. The necessity is obvious. Now we prove 

the sufficiency. Suppose by contrary that G is not 

supra β-connected. Then there exist two nonempty 

supra β-separated P,Q in X such that G = P 

Q. Choose x P and y O. So, x, y G and 

hence by hypothesis there exists a supra β- 

connected set H such that x, y H G. Thus H ∩ 

P and H ∩Q are non-empty supra β-separated sets 

with H = (P ∩ H) (Q ∩ H), a contradict to the 

supra β−connectedness of H. 

Theorem 3.9. If f : X → Y is a supra β-continuous 

surjective map and C, D are supra β-separated 

sets in Y, then f -1 (C), f -1 (D) are supra β- 

separated in X. 

Proof. Since f is surjective, then f -1 (C) and f -1 (D) 

are non-empty sets in X. Suppose by contrary that 

f -1 (C) and f -1 (D) are not supra β-separated sets in 

X. Then f -1 (C) ∩ f -1 (D)) ϕ. Since f is a 

supra β-continuous map, then by Theorem 1.1 we 

have f -1 (C)∩ f -1 (Cl(D)) ϕ. Thus C ∩ Cl(D) ϕ. 

Therefore C ∩ D) ϕ. Similarly, (C) ∩D 

ϕ. Thus C and D are not supra β-separated in Y ,


a contradiction. Hence f -1 (C) and f -1 (D) are supra 

β-separated in X. 

Theorem 3.10. If f : X→ Y is supra β-continuous 

bijective and A is supra β-connected in X, then 

f(A) is supra β-connected in Y . 

Proof. Suppose by contrary that f(A) is not supra 

β-connected in Y . Then f(A) = C D, where C 

and D are two non-empty supra β-separated in Y . 

By Theorem 3.9, we have f -1 (C) and f -1 (D) are 

not supra β-separated in X. Since f is bijective, 

then A = f -1 (f(A)) = f -1 (C) f -1 (D). 

Hence A is not supra β-connected in X, a 

contradiction. Thus f(A) is supra β-connected in Y . 

REFERENCES 

1. Devi, R., S. Sampathkumar & M. Caldas. On supra 

α-open sets and sα-continuous maps. General 

Mathematics 16 (2): 77-84 (2008). 

2. Mashhour, A.S., A.A. Allam, F.S. Mahmoud & 

F.H. Khedr. On supra topological spaces, Indian J. 

Pure and Appl. Math. 14 (4): 502-510 (1983). 

3. Ravi, O., G. Ramkumar & M. Kamaraj. On Supra 

β-open Sets and Supra β-continuity on Topological 

Spaces. Proceedings of UGC Sponsored National 

Seminar on Recent Developments in Pure and 

Applied Mathematics, 20-21 January 2011, 

Sivakasi, India. 

4. O.R. Sayed & T. Noiri. On supra b-open sets and 

supra b-continuity. Eur. J. Appl. Math. 3 (2): 295- 

302 (2010).

24 O.R. Sayed

Proceedings of the Pakistan Academy of Sciences 49 (1) 25-31 (2012) 


ISSN: 0377 - 2969 



A Study on Subordination Results for Certain Subclasses of 

Analytic Functions defined by Convolution 

M.K. Aouf*, A.A. Shamandy, A.O. Mostafa and A.K. Wagdy 


Mansoura University, Mansoura 35516, Egypt 

Abstract: In this paper, we drive several interesting subordination results of certain classes of analytic 

functions defined by convolution. 

Keywords and phrases: Analytic function, Hadamard product, subordination, factor sequence. 

2000 Mathematics Subject Classification: 30C45 


Let A denote the class of functions of the form: 

∞ 

f(z) = z + a k z k , 

k=2 

(1.1) 

which are analytic in the open unit disc U = 

{z ∈ C: |z| < 1}. Let φ ∈ A be given by 

∞ 

φ(z) = z + c k z k . (1.2) 

k=2 

Definition 1. (Hadamard product or convolution). 

Given two functions f and φ in the class A, 

where f(z) is given by (1.1) and φ(z) is given by 

(1.2) the Hadamard product (or convolution) 

f ∗ φ of f and φ is defined (as usual) by 

∞ 

(f ∗ φ)(z) = z + a k c k z k = (φ ∗ f)(z). (1.3 

k=2 

We also denote by K the class of functions 

f(z) ∈ A that are convex in U. 

Let M(β) be the subclass of A consisting of 

_____________________ 

functions f(z) which satisfy the inequality: 

Re zf′ (z) 

< β (z ∈ U), (1.4 

f(z) 

for some β > 1. Also let N(β) denote the 

subclasse of A consisting of functions f(z) which 

satisfy the inequality: 

Re 1 + zf′′(z) < β (z ∈ U), (1.5) 

f′(z) 

for some β > 1 ( see [7], [8], [9] and [10] ). For 

1 < β ≤ 4 , the classes M(β) and N(β) were 

3 

investigated earlier by Uralegaddi et al. [14] ( see 

also [12] and [13]). 

It follows from (1.4) and (1.5) that 

f(z) ∈ N(β) ⇔ zf ′ (z) ∈ M(β). (1.6) 

For 0 ≤ λ < 1, β > 1 and for all z ∈ U, let 

T(g, λ, β) be the subclass of A consisting of 

functions f(z) of the form (1.1) and functions 

g(z) given by: 

∞ 

g(z) = z + ∑k=2 b k z k (b k > 0), (1.7) 

which satisfying the analytic criterion: 

Received, September 2011; Accepted, March 2012 

*Corresponding author, M.K. Aouf; Email: mkaouf127@yahoo.com

26 Subordination Results for Certain Subclasses of Analytic Functions 

z(f ∗ g)′(z) 

Re 

< β. (1.8) 

(1 − λ)(f ∗ g)(z) + λz(f ∗ g)′(z) 

We note that: 

(i) T( z 

z 

, 0, β) = M(β) and T( , 0, β) 

1−z (1−z)² 

= N(β) (β > 1) (see [7] ); 

(ii) T(g, 0, β) = M(g, β)(β > 1)(see [1]). 

Also we note that: 

(i) T z , λ, β = T 1−z M(λ, β) 

zf′(z) 

= f ∈ A: Re 

(1 − λ)f(z) + λzf′(z) 

< β (0 ≤ λ < 1, β > 1, z ∈ U) ; 

z 

(ii) T , λ, β = T (1−z)² 

N(λ, β) 

= ∈ A: Re f′ (z) + zf ′′ (z) 

f ′ (z) + λzf ′′ (z) 

< β (0 ≤ λ < 1, β > 1, z ∈ U) ; 

∞ 

(iii) T z+ Γ k (α₁)z k 

k=2 

, λ, β = T q,s (α₁, λ, β) 

⎧ ⎧ 

⎪ ⎪ z(Hq,s (α 1 , β 1 )f (z))′ ⎪ ⎫ ⎫ 

⎪ 

= ∈ A: Re 

< , 

⎨ ⎨(1 − λ)H q,s (α 1 , β 1 )f (z) + 

⎪ ⎪ 

⎩ ⎩ λz(H q,s (α 1 , β 1 )f (z))′ ⎭ 

β⎭ 

⎪⎬ ⎬ 

⎪ 

where Γ k (α 1 ) is defined by 

Γ k (α 1 ) = 

(α 1 ) k−1 … . α q k−1 

(β 1 ) k−1 … . (β s ) k−1 (1) k−1 

(1.9) 

α i > 0, i = 1, . . . , q; β j > 0, j = 1, . . . , s; q 

≤ s + 1, q, s ∈ N 0 , N 0 

= N ∪ {0}, N = {1,2, . . }), 

and the operator H q,s (α 1 , β 1 ) was introduced and 

studied by Dziok and Srivastava ([4] and [5]), 

which is a generalization of many other linear 

operators considered earlier; 

∞ 

(iv)T z+ l+1+μ(k−1) m 

z k 

l+1 

k=2 

, λ, β = T(m, μ, l, λ, β) 

⎧ ⎧ z(I m ⎫ ⎫ 

(μ, l)f(z))′ 

= ∈ A: Re 

⎨ ⎨(1 − λ)I m < β , 

(μ, l)f(z) + ⎬ ⎬ 

⎩ ⎩ λz(I m (μ, l)f(z))′ ⎭ ⎭ 

where m ∈ N 0 , μ, l ≥ 0, z ∈ U and the 

operator I m (μ, l) was defined by Cătaş et al. [3], 

which is a generalization of many other linear 

operators considered earlier; 

∞ 

(v)T z+ C k (b,μ)z k 

k=2 

, λ, β = T(μ, b, λ, β) 

z(J μ 

= f ∈ A: Re 

b 

f(z))′ 

(1 − λ)J μ b f(z) + λz(J μ b f(z))′ 

Where C k (b, μ) is defined by 

< β (0 ≤ λ < 1, β > 1, z ∈ U), 

C k (b, μ) = 1 + b 

k + b μ 

(μ ∈ C, b ∈ C {Z₀−}, Z₀− = Z\N), (1.10) 

and the operator J b 

μ was introduced by Srivastava 

and Attiya [11], which is a generalization of many 

other linear operators considered earlier. 

Definition 2. (Subordination principle). For two 

functions f and φ, analytic in U, we say that the 

function f(z) is subordinate to φ(z) in U, 

written f (z) ≺ φ(z), if there exists a Schwarz 

function w (z), which (by definition) is analytic in 

U with w (0) = 0 and |w(z)| < 1, such that 

f(z) = φ(w(z)). Indeed it is known that 

f(z) ≺ φ(z) ⇒ f(0) 

= φ(0) and f(U) ⊂ φ(U ). 

Furthermore, if the function φ is univalent in U, 

then we have the following equivalence ( see [2] 

and [6] ): 

f(z) ≺ φ(z) ⇔ f(0) 

= φ(0)and f(U) ⊂ φ(U ). (1.11) 

Definition 3. ( Subordinating factor sequence ) 

[15]. A sequence {d k } ∞ 

k=1 of complex numbers is 

said to be a subordinating factor sequence if, 

whenever f of the form (1.1) is analytic, univalent

M.K. Aouf et al 27 

∞ 

∑ k=2(1 − λ)(k − 1) b k |a k | 

and convex in U, we have 

< 

2(β − 1) − ∑ 

∞ 

k=2 

|k − (2β − 1)[1 + λ(k − 1)]| b k |a k | < 1. 

∞ 

d k a k z k This completes the proof of Lemma 2. 

≺ f(z) (a 1 = 1; z ∈ U ). 

k=2 

Corollary 1. Let the function f(z) defined by 

(1.1) be in the class T(g; λ, β), then 


2(β − 1) 

|a k | ≤ 

. 

{(1 − λ)(k − 1) + |k − (2β − 1)[1 + λ(k − 1)]|}b 

Unless otherwise mentioned, we assume 

k 

(2.3) 

throughout this paper that 0 ≤ λ < 1, β > 1, z ∈ 

U and g(z) is given by (1.7) with b k+1 ≥ The result is sharp for the function 

b k (k ≥ 2). 

2(β − 1) 

f(z) = z + 

. 

{(1 − λ)(k − 1) + |k − (2β − 1)[1 + λ(k − 1)]|}b 

To prove our main result we need the following 

k 

lemmas. 

(2.4) 

∞ 

Lemma 1. [15]. The sequence {d k } k=1 is a 

Let T ∗ (g; λ, β) denote the subclass of functions 

subordinating factor sequence if and only if 

f(z) ∈ A whose coefficients satisfy the condition 

∞ 

(2.2). We note that T ∗ (g; λ, β) ⊆ T(g, λ, β). 

Re 1 + 2 d k z k > 0. (2.1) 

Thereom 1. Let f(z) ∈ T ∗ (g; λ, β). Then 

k=1 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

Now, we prove the following lemma which gives 

(f ∗ h)(z) ≺ h(z), 

2{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } 

a sufficient condition for functions belonging to 

(2.5) 

the class T(g, λ, β): 

for every function h ∈ K, and 

Lemma 2. A function f(z) of the form (1.1) is Re{f(z)} > − {2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } 

. 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 

said to be in the class T(g, λ, β) if 

2 

(2.6) 

(1 − λ)(k − 1) 

[1−λ+|3−2β−λ(2β−1)|]b 

∞ 

∑k=2 k − (2β − 1) 

+ 

[1 + λ(k − 1)] b k |a k | ≤ 2(β − 1). (2.2) 

The constant factor 

2 

2{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]b 2 } 

in the subordination result (2.5) is the best 

estimate. 

Proof. Assume that the inequality (2.2) holds true. 

Then it suffices to show that 

Proof. Let f(z) ∈ T ∗ (g; λ, β) and suppose that 

∞ 

z(f ∗ g)′(z) 

(1 − λ)(f ∗ g)(z) + λz(f ∗ g)′(z) − 1 

h(z) = z + ∑k=2 h k z k ∈ K, then 

 

< 1. 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 

z(f ∗ g)′(z) 

(1 − λ)(f ∗ g)(z) + λz(f ∗ g)′(z) − (2β − 1) 2 

(f ∗ h)(z) 

2{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } 

∞ 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

We have 

= 

2{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } z + h k a k z k . 

k=2 

(2.7) 

z(f ∗ g)′(z) 

(1 − λ)(f ∗ g)(z) + λz(f ∗ g)′(z) − 1 Thus, by using Definition 3, the subordination 

 

result holds true if 

z(f ∗ g)′(z) 

(1 − λ)(f ∗ g)(z) + λz(f ∗ g)′(z) − (2β − 1) ∞ 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

 

2{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 

∞ 

∑ k=2(1 − λ)(k − 1) b k |a k ||z| k−1 

2 } a k 

k=1 

≤ 

2(β − 1) − ∑ 

∞ 

k=2 

|k − (2β − 1)[1 + λ(k − 1)]| b k |a k ||z| k−1


is a subordinating factor sequence, with a 1 = 1. 

In view of Lemma 1, this is equivalent to the 

following inequality: 

∞ 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

Re 1 + 

{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } a k z k > 0. 

k=1 

Now, since 

Ψ(k) = {(1 − λ)(k − 1) + |k 

−(2β − 1)[1 + λ(k − 1)]|}b k 

(2.8) 

is an increasing function of k (k ≥ 2), we have 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

Re 1 + 

{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } a k z k 

∞ 

k=1 

[1−λ+|3−2β−λ(2β−1)|]b 

= Re 1 + 

2 

{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]b 2 } z 

+ ∑ ∞ k=2 [1−λ+|3−2β−λ(2β−1)|]b 2a k z k 

 

{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]b 2 } 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

≥ 1 − 

{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } r 

1 

− 

[1 

{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]b 2 } 

k=2 

− λ + |3 − 2β − λ(2β − 1)|]b 2 |a k |r k 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

≥ 1 − 

{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } r 

− 

1 

{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]b 2 } ∙ 

∞ 

∙ {(1 − λ)(k − 1) 

k=2 

+ |k − (2β − 1)[1 + λ(k − 1)]|} b k |a k |r k 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

≥ 1 − 

{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } r 

2(β − 1) 

− 

{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } r 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

≥ 1 − 

{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } 

2(β − 1) 

− 

{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } 

> 0 (|z| = r < 1), 

where we have also made use of assertion (2.2) of 

Lemma 2. Thus (2.8) holds true in U. This proves 

the inequality (2.5). The inequality (2.6) follows 

from (2.5) by taking the convex function 

z 

h(z) = = z + ∑∞ 

1−z k=2 zk ∈ K. (2.9) 

To prove the sharpness of the constant 

∞ 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

2{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } , 

we consider the function f 0 (z) ∈ T ∗ (g; λ, β) 

given by 

2(β − 1) 

f 0 (z) = z − 

z 2 . 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

Thus from (2.5), we have 

[1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

2{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } f 0 (z) ≺ z 

1 − z 

It is easily verified that 

min Re [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 

|z|≤r 2{2(β − 1) + [1 − λ + |3 − 2β − λ(2β − 1)|]b 2 } f 0 (z) 

= − 1 2 . 

This show that the constant 

[1−λ+|3−2β−λ(2β−1)|]b 2 

is the best 

2{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]b 2 } 

possible. This completes the proof of Theorem 1. 

Remark. (i) Taking g(z) = 

z and λ = 0 in 

1−z 

Lemma 2 and Theorem 1, we obtain the result 

obtained by Srivastava and Attiya [10, Corollary 

2] and Nishiwaki and Owa [7, Theorem 2.1]; 

(ii) Taking g(z) = 

z 

(1−z) 

2 

and λ = 0 in Lemma 

2 and Theorem 1, we obtain the result obtained by 

Srivastava and Attiya [10, Corollary 4] and 

Nishiwaki and Owa [7, Corollary 2.2]. 

Also, we establish subordination results for the 

associated subclasses, M ∗ (g, β), T ∗ M (λ, β), 

T ∗ N (λ, β), T ∗ q,s (α 1 , λ, β), T ∗ (m, μ, l, λ, β) and 

T ∗ (μ, b, λ, β), whose coefficients satisfy the 

condition (2.2) in the special cases as mentioned 

in the introduction. 

By taking λ = 0 in Lemma 2 and Theorem 1, we 

obtain the following corollary: 


(1.1) be in the class M ∗ (g, β) and satisfy the 

condition 

∞ 

{k − 1 + |k − (2β − 1)|} b k |a k | ≤ 2(β − 1). (2.11) 

k=2 

Then for every function h ∈ K, we have:


[1 + |3 − 2β|]b 2 

(f ∗ h)(z) ≺ h(z) 

2{2(β − 1) + (1 + |3 − 2β|)b 2 } 

(2.12) 

and 

Re{f(z)} > − {2(β − 1) + (1 + |3 − 2β|)b 2 } 

[1 + |3 − 2β|]b 2 

. 

(2.13) 

[1+|3−2β|]b 


2 

in the 

2{2(β−1)+(1+|3−2β|)b 2 } 

subordination result (2.12) can not be replaced by 

a larger one and the function 

f 0 (z) = z − 

gives the sharpness. 

2(β − 1) 

[1 + |3 − 2β|]b 2 

z 2 (2.14) 

By taking g(z) = 

z in Lemma 2 and Theorem 

1−z 

1, we obtain the following corollary: 


(1.1) be in the class T M ∗ (λ, β) and satisfy the 

condition 

∞ 

(1 − λ)(k − 1) + 

 

|k − (2β − 1)[1 + λ(k − 1)]| |a k | ≤ 2(β − 1). 

k=2 

Then for every function h ∈ K, we have: 

(2.15) 

[1 − λ + |3 − 2β − λ(2β − 1)|] 

(f ∗ h)(z) ≺ h(z) 

2[2β − 1 − λ + |3 − 2β − λ(2β − 1)|] 

(2.16) 

and 

Re{f(z)} > − 


[2β − 1 − λ + |3 − 2β − λ(2β − 1)|] 

. 

[1 − λ + |3 − 2β − λ(2β − 1)|] 

(2.17) 

[1−λ+|3−2β−λ(2β−1)|] 

2[2β−1−λ+|3−2β−λ(2β−1)|] in 

the subordination result (2.16) can not be 

replaced by a larger one and the function 

2(β − 1) 

f 0 (z) = z − 

[1 − λ + |3 − 2β − λ(2β − 1)|] z2 


(2.18) 

z 

(1−z) 

By taking g(z) = in Lemma 2 and 

2 

Theorem 1, we obtain the following corollary: 


(1.1) be in the class T N ∗ (λ, β) and satisfy the 

condition 

∞ 

(1 − λ)(k − 1) + 

k 

|k − (2β − 1)[1 + λ(k − 1)]| |a k | ≤ 2(β − 1). 

k=2 


(2.19) 

[1 − λ + |3 − 2β − λ(2β − 1)|] 

(f ∗ h)(z) ≺ h(z) 

2[β − λ + |3 − 2β − λ(2β − 1)|] 

(2.20) 

and 

[β − λ + |3 − 2β − λ(2β − 1)|] 

Re{f(z)} > − 

[1 − λ + |3 − 2β − λ(2β − 1)|] . 

(2.21) 

The constant factor [1−λ+|3−2β−λ(2β−1)|] 

in the 

2[β−λ+|3−2β−λ(2β−1)|] 



β − 1 

f 0 (z) = z − 

[1 − λ + |3 − 2β − λ(2β − 1)|] z2 


(2.22) 

By taking b k = Γ k (α 1 ) , where Γ k (α 1 ) defined 

by (1.9), in Lemma 2 and Theorem 1, we obtain 

the following corollary: 


(1.1) be in the class T ∗ q,s (α 1 , λ, β) and satisfy the 

condition 

∞ (1 − λ)(k − 1) 

k − (2β − 1) 

+ 

[1 + λ(k − 1)] Γ k (α 1 )|a k | ≤ 2(β − 1). 

k=2 


(2.23) 

[1 − λ + |3 − 2β − λ(2β − 1)|]Γ 2 (α 1 ) 

(f ∗ h)(z) ≺ h(z) 

2(β − 1) 

2 

+[1 − λ + |3 − 2β − λ(2β − 1)|]Γ 2 (α 1 ) 

(2.24)


and 

2(β − 1) 

 

+ 1 − λ + 3 − 2β − 

λ(2β − 1) Γ 2(α 1 ) 

Re{f(z)} > − 

[1 − λ + |3 − 2β − λ(2β − 1)|]Γ 2 (α 1 ) . 


[1−λ+|3−2β−λ(2β−1)|]Γ 2 (α 1 ) 

2{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]Γ 2 (α 1 )} 

in the 

(2.25) 



2(β − 1) 

f 0 (z) = z − 

[1 − λ + |3 − 2β − λ(2β − 1)|]Γ 2 (α 1 ) z2 


(2.26) 

By taking b k = l+1+μ(k−1) 

m (m ∈ N 

l+1 

0 , μ, l ≥ 

0) in Lemma 2 and Theorem 1, we obtain the 

following corollary: 


(1.1) be in the class T ∗ (m, μ, l, λ, β) and satisfy 

the condition 

∞ (1 − λ)(k − 1) 

l + 1 + μ(k − 1) 

k − (2β − 1) 

+ 

[1 + λ(k − 1)] 

l + 1 

k=2 


m 

|a k | ≤ 2(β − 1). 

(2.27) 

[1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m 

(f ∗ h)(z) ≺ h(z) 

2(l + 1) m (β − 1) 

2 

+ 1 − λ + 3 − 2β − 

λ(2β − 1) (l + 1 + μ)m 

and 

(2.28) 

2(l + 1) m (β − 1) 

 

3 − 2β 

+ 1 − λ + 

(l + 1 + μ)m 

−λ(2β − 1) 

Re{f(z)} > − 

[1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m. 

(2.29) 


[1−λ+|3−2β−λ(2β−1)|](l+1+μ) m 

2{2(l+1) m (β−1)+[1−λ+|3−2β−λ(2β−1)|](l+1+μ) m } 

in the subordination result (2.28) can not be 

replaced by a larger one and the function 

2(β − 1)(l + 1) m 

f 0 (z) = z − 

[1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m z2 

(2.30) 


By taking b k = C k (b, μ), where C k (b, μ) 

defined by (1.10), in Lemma 2 and Theorem 1, we 

obtain the following corollary: 


(1.1) be in the class T ∗ (μ, b, λ, β) and satisfy the 

condition 

∞ (1 − λ)(k − 1) 

k − (2β − 1) 

+ 

[1 + λ(k − 1)] C k (b, μ)|a k | ≤ 2(β − 1). 

k=2 


(2.31) 

[1 − λ + |3 − 2β − λ(2β − 1)|]C 2 (b, μ) 

(f ∗ h)(z) ≺ h(z) 

2(β − 1) 

2 

3 − 2β 

+ 1 − λ + 

−λ(2β − 1) C 2 (b, μ) 

and 

(2.32) 

2(β − 1) 

 

3 − 2β 

+ 1 − λ + 

−λ(2β − 1) C 2(b, μ) 

Re{f(z)} > − 

1 − λ + 3 − 2β − 

. 

λ(2β − 1) C 2(b, μ) 


[1−λ+|3−2β−λ(2β−1)|]C 2 (b,μ) 

2{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]C 2 (b,μ)} 

in the 

(2.33) 



2(β − 1) 

f 0 (z) = z − 

[1 − λ + |3 − 2β − λ(2β − 1)|]C 2 (b, μ) z2 


3. ACKNOWLEDGEMENTS 

(2.34) 

The authors thank the anonymous referees of the paper 

for their helpful suggestions. 

4. REFERENCES 

1. Aouf, M.K., A.A.. Shamandy, A.O. Mostafa & 

E.A. Adwan. Subordination results for certain 

class of analytic functions defined by convolution.


Rend. del Circolo Math. di Palermo (in press). 

2. Bulboaca, T. Differential Subordinations and 

Superordinations, Recent Results. House of 

Scientific Book Publ., Cluj-Napoca (2005). 

3. A Cătaş, G.I. Oros & G. Oros. Differential 

subordinations associated with multiplier 

transformations. Abstract Appl. Anal. ID845724: 

1-11 (2008). 

4. Dziok, J. & H.M. Srivastava. Classes of analytic 

functions with the generalized hypergeometric 

function. Appl. Math. Comput. 103: 1-13 (1999). 

5. Dziok, J. & , H.M. Srivastava. Certain subclasses 

of analytic functions associated with the 

generalized hypergeometric function. Integral 

Transform. Spec. Funct. 14: 7-18 (2003). 


Subordinations Theory and Applications. In: Series 

on Monographs and Textbooks in Pure and 

Applied Mathematics 255. Marcel Dekker, New 

York (2000). 

7. Nishiwaki, J. & S Owa.. Coefficient inequalities 

for certain analytic functions, Internat. J. Math. 

Math. Sci. 29 (5): 285-290 (2002). 

8. Owa, S. & J. Nishiwaki. Coeffocient estimates for 

certain classes of analytic functions. J. Inequal. 

Pure Appl. Math. 3 (5), Art. 72: 1-12 (2002). 

9. Owa, S. & H.M. Srivastava. Some generalized 

convolution properties associated with certain 

subclasses of analytic functions. J. Inequal. Pure 

Appl. Math. 3 (3), Art. 42: 1-27 (2002). 

10. Srivastava, H.M. & A.A Attiya. Some 

subordination results associated with certain 

subclasses of analytic functions. J. Inequal. Pure 

Appl. Math. 5 (4), Art. 82: 1-14 (2004). 

11. Srivastava, H.M. & A.A Attiya.. An integral 

operator associated with the Hurwitz-Lerch Zeta 

function and differential subordination. Integral 

Transform. Spec. Funct. 18: 207-216 (2007). 

12. Srivastava, H.M. & S. Owa. Current Topics in 

Analytic Function Theory. World Scientific 

Publishing Company, Singapore (1992). 

13. Uralegaddi, BA. & A.R. Desa. Convolutions of 

univalent functions with positive coefficients. 

Tamkang J. Math. 29: 279-285 (1998). 

14. Uralegaddi, B.A., M.D. Ganigi & S.M Sarangi. 

Univalent functions with positive coefficients. 

Tamkang J. Math. 25: 225-230 (1994). 

15. Wilf, S. Subordinating factor sequence for convex 

maps of the unit circle. Proc. Amer. Math. Soc. 12: 

689-693 (1961).



ISSN: 0377 - 2969 



Existence and Uniqueness for Solution of Differential Equation 

with Mixture of Integer and Fractional Derivative 

Shayma Adil Murad 1 , Rabha W. Ibrahim 2,* and Samir B. Hadid 3 

1 Department of Mathematics, Faculty of Science, Duhok University, Kurdistan, Iraq 

2 Institute of Mathematical Sciences, University Malaya, 50603, Malaysia 

3 Department of Mathematics and Basic Sciences, College of Education and Basic 

Sciences, Ajman University of Science and Technology, UAE 

Abstract: By employing the Krasnosel’ski˘ıfixed point theorem, we establish the existence of solutions for 

mixed differential equation (ordinary and fractional ). Moreover, we suggest the uniqueness of solution 

and we examine our abstract results by applications. 

Keywords: Fractional calculus, Fractional differential equation, Integral boundary condition, 

Krasnosel’skiĭ Fixed Point Theorem 


In recent years, fractional equations have gained 

considerable interest due to their applications in 

various fields of the science such as physics, 

mechanics, chemistry, biology, engineering and 

computer sciences. Significant development has 

been made in ordinary and partial differential 

equations involving fractional derivatives [1, 2]. 

The class of fractional differential equations of 

various types plays an important role not only in 

mathematics but also in physics, control systems, 

diffusion, dynamical systems and engineering to 

create the mathematical modeling of many 

physical phenomena. 

The existence of positive solution and multipositive 

solutions for nonlinear fractional. 

Moreover, by using the concepts of the 

subordination and superordination of analytic 

functions, the existence of analytic solutions for 

fractional differential equations in complex 

domain are posed in [3, 4]. About the development 

of existence theorems for fractional functional 

differential equations. 

Many papers on fractional differential 

equations are devoted to existence and uniqueness 

of solutions such a type of equations (e.g., [5, 6]). 

In this paper we investigate the existence of 

solution of differential equation with mixture of 

integer and fractional derivative. Our result is an 

application of Krasnosel’skiĭ fixed point theorem. 

Such differential equation plays a very important 

rule in applications in sciences and engineering 

problems [7]. 

2. PRELIMINARIES 

Recall the following basic definitions and results: 

Definition 2.1. For a function f given on the 

interval [a,b], the Caputo fractional order 

derivative off is defined by 

where and denote the integer part 

of 

_____________________ 

Received, October 2011; Accepted, March 2012 

*Corresponding author: Rabha W. Ibrahim, E-mail: rabhaibrahim@yahoo.com

34 Shayma Adil Murad et al 

Lemma 2.2. Let 

0, then 

for somec i 

R , i=0,1,..,n-1 , n [ ] 1 

Definition 2.3. Let f be a function which is 

defined almost everywhere on [a,b] , for 0 , 

we define 

b 

a 

b 

1 

1 

( ) ( ) ( ) 

 

I f t b f d 

( ) 

 

a 

provided that the integral (Lebesgue) exists. 

Theorem 2.4. [8] (Krasnosel’skiĭ Theorem) 

Let M be a closed convex bounded nonempty 

subset of a Banach space X. Let A and B be 

two operators such that: 

i)Ax+By=M, whenever x, y M ; 

ii) A is compact and continuous ; 

iii) B is a contraction mapping . 

Then, there exists 

such thatz =Az+Bz. 

Let R be a Banach space with the norm . 

Let 

, be Banach space of all 

continuous functions , with 

supermum norm 

. 

Consider the extraordinary differential 

equation with initial conditions , which has the 

form 

Proof. we reduce the problem (2.1) to an 

equivalent integral equation 

c 

(2.3) 

Operate both side of equation (2.3) by the operator 

, we get 

c 

 

In view of the relations 

, for , and by 

Lemma (2.2), we obtain 

c 

 

c 

 

c 

 

By applying the condition (2.2), we get 

and 

c 

 

c 

 

(2.4) 

Then by substitute and in equation (2.4) , we 

get 

(2.1) 

and 

(2.2) 

Where is the Caputo fractional derivative and 

the nonlinear functions 

is 

continuous . 

c 

 

c 

 

The equation (2.5) will become of the form 

(2.5) 

Lemma 2.5. Let and 

be a continuous function, then the solution of 

fractional differential equation (2.1) with the 

initial condition (2.2) is : 

which completes the proof.

Existence and Uniqueness for Solution of Differential Equation 35 

To prove the main results, we need the following 

assumptions: 

(H1) There exists 

such that 

for and . 

(H2)There exists constants 

such that 

, . 

(H3) 

, for all 

and 

For convenience,let us set 

(2.6) 


Theorem 3.1. Assume that 

is a 

continuous function and satisfies the assumption 

(H1).Then the boundary value problem(2.1) has a 

unique solution. 

Proof. Consider the operator T:C C by 

Setting 

For y 

, we show that 

, where . 

, we have 

Where 

, we obtain: 

Now, for and for each ,we 

obtain:

36 Shayma Adil Murad et al 

where 

As , therefore is a contraction. Thus, the 

conclusion of the theorem follows by the 

contraction mapping principle (Banach fixed point 

theorem). 

Theorem 3.2. Let ∶ [0,b] × R → R be a 

continuous function mapping bounded subsets 

of [0,b] × R into relatively compact subsets of R, 

and the assumptions ( H2) and ( H3) hold. Then 

the boundary value problem (2.1) has at least one 

solution on [0,b]. 

It is clear that is contraction mapping, 

Continuity of implies that the operator is 

continuous. Also, is uniformly bounded on 

as 

Now we prove the compactness of the operator . 

We define , 

And consequently we have 

Proof. Letting and 

consider 

. We define the 

operators and s 

for 

, we find that 

Which is independent of x. Thus, is 

equicontinuous. Using the fact that maps 

bounded subset into relatively compact subsets, so 

is relatively compact on . Hence, by the 

Arzelá-Ascoli Theorem, is compact on . Thus 

all the assumptions of Theorem 3.2 aresatisfied. 

So the conclusion of Theorem 3.2 implies that the 

initial value problem (2.1) has at least one solution 

on [0, b]. 

4. REFERENCES 

Now prove that 

is contraction mapping 

1. Kilbas, A.A., H.M. Srivastava & J.J. Trujillo. 

Theory and Applications of Fractional 

Differential Equations. North-Holland 

Mathematics Studies Vol. 204. Elsevier Science 

B.V., Amsterdam, The Netherlands, p. 347-463 

(2006). 

2. Podlubny, I. Fractional Differential Equations, 

Vol. 198 of Mathematics in Science and 

Engineering. Academic Press, San Diego, CA, 

USA, p. 261-307 (1999).

Existence and Uniqueness for Solution of Differential Equation 37 

3. Ibrahim, R.W. & M. Darus. Subordinati-on and 

superordination for analytic functions involving 

fractional integral operator. Complex Variables 

and Elliptic Equations 53: 1021-1031 (2008). 

4. Ibrahim, R.W. & M. Darus. Subordi-nation and 

superordination for univalent solutions for 

fractional differential equations. J. Math. Anal. 

Appl. 345: 871-879 (2008). 

5. Hadid S.B. Local and global existence theorems 

on differential equations of non-integer order. J. 

Fractional Calculus 7: 101-105 (1995). 

6. Murad, S.M. & H.J. Zekri & S. Hadid. Existence 

and uniqueness theorem of fractional mixed 

Volterra-Fredholm integro-differential equation 

with integral boundary conditions. International 

Journal of Differential Equations 2011: 1-15 

(2011). 

7. Cieielski, M. & J.S. Leszczynski. Numerical 

solution to boundary value problem for 

anomalous diffusion equation with Riesz-Feller 

fractional operator. J. Theo. and Appl. Mech. 44: 

393-403 (2006). 

8. Krasnosel’skiĭ, M.A. Two remarks on the method 

of successive Approximati-ons. Uspekhi 

Matematicheskikh Nauk 63: 123–127 (1955).

34 Shayma Adil Murad et al



ISSN: 0377 - 2969 



On Stability of a Class of Fractional Differential Equations 

Rabha W. Ibrahim* 

Institute of Mathematical Sciences, University Malaya, 

Kuala Lumpur 50603, Malaysia 

Abstract: In this paper, we consider the Hyers-Ulam stability for fractional differential equations of the 

 

form: D z 

f ( z) 

= G( 

f ( z), 

zf ( 

z); 

z), 

1< 2 in a complex Banach space. Furthermore, 

applications are illustrated. 

Keywords: Analytic function; Unit disk; Hyers-Ulam stability; Admissible functions; Fractional calculus; 

Fractional differential equation 


A classical problem in the theory of functional 

equations is that: If a function f approximately 

satisfies functional equation E when does there 

exists an exact solution of E which f 

approximates. In 1940, Ulam [1] imposed the 

question of the stability of Cauchy equation and 

in 1941, D. H. Hyers solved it [2]. In 1978, 

Rassias [3] provided a generalization of Hyers, 

theorem by proving the existence of unique linear 

mappings near approximate additive mappings. 

The problem has been considered for many 

different types of spaces [4-6]. Recently, Li and 

Hua [7] discussed and proved the Hyers-Ulam 

stability of spacial type of finite polynomial 

equation, and Bidkham, Mezerji and Gordji [8] 

introduced the Hyers-Ulam stability of 

generalized finite polynomial equation. Finally, 

Rassias [9] imposed a Cauchy type additive 

functional equation and investigated the 

generalised Hyers-Ulam “product-sum” stability 

of this equation. 

The class of fractional differential equations 

of various types plays important roles and tools 

not only in mathematics but also in physics, 

control systems, dynamical systems and 

engineering to create the mathematical modeling 

of many physical phenomena. Naturally, such 

_____________________ 

equations required to be solved. Many studies on 

fractional calculus and fractional differential 

equations, involving different operators such as 

Riemann-Liouville operators [10], Erdélyi-Kober 

operators [11], Weyl-Riesz operators [12], 

Caputo operators [13] and Grünwald-Letnikov 

operators [14], have appeared during the past 

three decades. The existence of positive solution 

and multi-positive solutions for nonlinear 

fractional differential equation are established 

and studied [15]. Moreover, by using the 

concepts of the subordination and superordination 

of analytic functions, the existence of analytic 

solutions for fractional differential equations in 

complex domain are suggested and posed [16- 

18]. 

Srivastava and Owa [19] gave definitions for 

fractional operators (derivative and integral) in 

the complex z-plane C as follows: 

1.1. Definition: The fractional derivative of order 

is defined, for a function f (z) 

by 

D 

 

z 

1 d 

f ( z) := 

(1) 

dz 

 

0 

z 

f ( ) 

( z 

) 

 

d 

, 

where the function f (z) 

is analytic in simplyconnected 

region of the complex z-plane C 

containing the origin and the multiplicity of 


*Email: rabhaibrahim@yahoo.com

40 Rabha W. Ibrahim 

(z ) 

is removed by requiring ( z 

) 

be real when ( z ) > 0. 

log to 

1.2. Definition: The fractional integral of order 

> 0 is defined, for a function f (z), 

by 

I 

 

z 

1 z 

f ( z) := f ( )( z 

) 

( 

) 

0 

1 

d 

; > 0, 

where the function f (z) 

is analytic in simplyconnected 

region of the complex z-plane (C) 

containing the origin and the multiplicity of 

1 

( z 

) is removed by requiring log ( z 

) 

to be real when ( z ) > 0. 

1.1. Remark: 

 

D z 

and 

 

I z 

z 

z 

 

 

( 

1) 

 

= z , > 1 

( 

 

1) 

( 

1) 

 

= z , > 1. 

( 

 

1) 

In [17], it was shown the relation 

I 

 

z 

D 

 

z 

 

f ( z) 

= D I f ( z) 

= f ( z). 

z 

z 

Let U := { z C:| 

z |< 1} be the open unit 

disk in the complex plane C and H denote the 

space of all analytic functions on U . Here we 

suppose that H as a topological vector space 

endowed with the topology of uniform 

convergence over compact subsets of U . Also 

for a C 

and m N, let H [ a, 

m] 

be the 

subspace of H consisting of functions of the 

form 

f ( z) 

= a a z 

m 

m 

a 

m1 

1z 

, z U. 

m 

Definition 1.3. Let p be a real number. We say 

that 

 

a z 

n 

n 

= f ( z) 

(1) 

n=0 

has the generalized Hyers-Ulam stability if there 

exists a constant K > 0 with the following 

property: 

 

for every > 0, wU 

= U U, 

if 

p 

| an 

| 

| | ( ), 

2 

pn ( 1) 

 

 

n 

aw 

n 

 

n=0 n=0 

p (0,1) 

then there exists some 

equation (1) such that 

i i 

| z w | K, 

( z , w U, 

i N). 

z U that satisfies 

In the present paper, we study the generalized 

Hyers-Ulam stability for holomorphic solutions 

of the fractional differential equation in complex 

Banach spaces X and Y 

 

D z 

f ( z) 

= G( 

f ( z), 

zf ( 

z); 

z), 

1 < 2, (2) 

2 

where G : X U 

Y 

and f : U X are 

holomorphic functions such that f (0) = ( 

is the zero vector in X ). 

Recently, the authors studied the ulam 

stability for different types of fractional 

differential equations [20-22]. 

2. RESULTS 

In this section we present extensions of the 

generalized Hyers-Ulam stability to holomorphic 

vector-valued functions. Let X , Y represent 

complex Banach space. The class of admissible 

functions G ( X , Y), 

consists of those functions 

2 

g : X U 

Y 

that satisfy the admissibility 

conditions: 

g( r, ks; z) 1, 

when r = 1, s = 1, 

( z U, 

k 1). 

We need the following results: 

(3) 

2.1. Lemma: [23] Let g G( X, 

Y). 

If 

f : U X is the holomorphic vector-valued 

functions defined in the unit disk U with 

f (0) = , 

then

On Stability of a Class of Fractional Differential Equations 41 

 

g( f ( z), zf ( z); z) < 1 

f( z) < 1. 

(4) 

2.1. Theorem: In Eq. (2), if G G( X, 

Y) 

is the 

holomorphic vector-valued function defined in 

the unit disk U then 

 

G( f ( z), zf ( z); z)

42 Rabha W. Ibrahim 

G( r, ks; z) = a( r k s ) 

b z a k b z 

2 n 2 

| | = (1 ) | | 1, 

when r = s =1, z U. 

Hence by 

Theorem 2.1, we have : If a 0.5, b 0 and 

f : U X is a holomorphic vector-valued 

function defined in U , with f (0) = , 

then 

a( f ( z) zf ( z) ) 

b z 

2 

| | < 1 f ( z) < 1. 

Consequently, I G( f ( z), zf ( z); z) 0, 

( 

z) 

for every z U. 

Consider the function 

G : X 

2 Y 

by 

s 

G( 

r, 

s; 

z) 

= r , 

( 

z) 

with G ( , 

) 

= . 

Now for r = s =1, 

we have 

k 

G( r, ks; z) =|1 | 1, 

() 

z 

k 1 

and thus G G( X, 

Y). 

If f : U X is a 

holomorphic vector-valued function defined in 

U , with f (0) = , 

then 

 

zf ( z) 

f( z) < 1 

( z) 

f( z) < 1. 

Hence, according to Theorem 2.2, f has the 

generalized Hyers-Ulam stability.

On Stability of a Class of Fractional Differential Equations 43 

4. REFERENCES 

1. Ulam, S.M. A Collection of Mathematical 

Problems. Interscience Publ. New York, 1961. 

Problems in Modern Mathematics. Wiley, New 

York (1964). 

2. Hyers, D.H. On the stability of linear functional 

equation. Proc. Nat. Acad. Sci. 27: 222-224 

(1941). 

3. Rassias, Th.M. On the stability of the linear 

mapping in Banach space. Proc. Amer. Math. Soc. 

72: 297-300 (1978). 

4. Hyers, D.H. The stability of homomorphisms and 

related topics, in Global Analysis-Analysis on 

Manifolds. Teubner-Texte Math. 75: 140-153 

(1983). 

5. Hyers, D.H. & Th.M. Rassias. Approximate 

homomorphisms. Aequationes Math. 44: 125-153 

(1992). 

6. Hyers, D.H., G. I. Isac & Th.M. Rassias. Stability 

of Functional Equations in Several Variables. 

Birkhauser, Basel (1998). 

7. Li, Y. & L. Hua. Hyers-Ulam stability of a 

polynomial equation. Banach J. Math. Anal. 3: 

86-90 (2009). 

8. Bidkham, M. & H.A. Mezerji & M.E. Gordji. 

Hyers-Ulam stability of polynomial equations. 

Abstract and Applied Analysis doi:10.1155/2010/ 

754120 (2010). 

9. Rassias, M.J. Generalised Hyers-Ulam “productsum” 

stability of a Cauchy type additive 

functional equation. European J. Pure and Appl. 

Math. 4: 50-58 (2011). 

10. Diethelm, K. & N. Ford. Analysis of fractional 

differential equations. J. Math. Anal. Appl. 265: 

229-248 (2002). 

11. Ibrahim, R.W. & S. Momani. On the existence 

and uniqueness of solutions of a class of fractional 

differential equations. J. Math. Anal. Appl. 334: 1- 

10 (2007). 

12. Momani, S.M. & R.W. Ibrahim. On a fractional 

integral equation of periodic functions involving 

Weyl-Riesz operator in Banach algebras. J. Math. 

Anal. Appl. 339: 1210-1219 (2008). 

13. Bonilla, B., M. Rivero & J.J. Trujillo. On systems 

of linear fractional differential equations with 

constant coefficients. App. Math. Comp. 187: 68- 

78 (2007). 

14. Podlubny, I. Fractional Differential Equations. 

Academic Press, London, (1999). 

15. Zhang, S. The existence of a positive solution for 

a nonlinear fractional differential equation. J. 

Math. Anal. Appl. 252: 804-812 (2000). 

16. Ibrahim, R.W. & M. Darus. Subordination and 

superordination for analytic functions involving 

fractional integral operator. Complex Variables 

and Elliptic Equations 53:1021-1031 (2008). 

17. Ibrahim, R.W. & M. Darus. Subordination and 

superordination for univalent solutions for 

fractional differential equations. J. Math. Anal. 

Appl. 345: 871-879 (2008). 

18 Ibrahim, R.W. Existence and uniqueness of 

holomorphic solutions for fractional Cauchy 

problem. J. Math. Anal. Appl. 380: 232-240 

(2011). 

19 Srivastava, H.M. & S. Owa. Univalent Functions, 

Fractional Calculus, and Their Applications. 

Halsted Press, John Wiley and Sons, New York 

(1989). 

20. Ibrahim, R.W. Generalized Ulam–Hyers stability 

for fractional differential equations. International 

Journal of Mathematics 23: 1-9 (2012). 

21. Ibrahim, R.W. On generalized Hyers-Ulam 

stability of admissible functions. Abstract and 

Applied Analysis (in press). 

22. Ibrahim, R.W. Approximate solutions for 

fractional differential equation in the unit disk. 

Electronic Journal of Qualitative Theory of 

Differential Equations 64: 1-11 (2011). 

23 Miller, S.S. & P.T. Mocanu. Differential 

Subordinantions: Theory and Applications. Pure 

and Applied Mathematics No. 225. Dekker, New 

York (2000).



ISSN: 0377 - 2969 



Some Inclusion Properties of p-Valent Meromorphic Functions 

defined by the Wright Generalized Hypergeometric Function 

M.K. Aouf, A.O. Mostafa, A.M. Shahin and S.M. Madian* 


Mansoura University, Mansoura 35516, Egypt 

Abstract: In this paper, using the Wright generalized hypergeometric function we define a new operator 

and some classes of meromorphic functions associated to it and investigate several inclusion properties of 

these classes. Some applications involving integral operator are also considered. 

Keywords and phrases: p-Valent meromorphic functions, Hadamard product, Wright generalized hypergeaometric 

hypergeaometric function, inclusion relationships 

2000 Mathematics Subject Classification : 30C45 


Let 

 

p 

denote the class of functions of the form: 

 

fz z p 

k1p 

a k z k p 1,2,..., 1.1 

(1.1) 

which are analytic and p-valent in the punctured 

unit disc U z : z and 0 | z| 1} U \{0} 

. For two analytic functions f and g in U , f 

is said to be subordinate to g, written f g or 

fz gz , if there exists an analytic function 

wz in U , with w0 0 and |wz| 1 

such that fz gwz . If gz is univalent 

function, then f g if and only if (see [8] and 

[16]) 

 

f 0 g 0 and f U g U . 

For functions fz p given by (1.1) and 

gz p defined by 

 

gz z p 

k1p 

b k z k p , 1.2 

(1.2) 

the Hadamard product (or convolution) of fz 

and gz is given by 

( f g)( z) 

z 

 

k 

a b z ( g f )( z). 

k 

1 p 

k 

k 

p 

(1.3) 

For 0 , p, let MS ,p, MK,p, 

MC,,p 

 

and MC ( , , p), 

be the 

subclasses of p consisting of all meromorphic 

functions which are, respectively, starlike of order 

, convex of order , close-to-convex of order 

and type and quasi-convex functions of 

order and type in U . Let S be the class 

of all functions which are analytic and 

univalent in U and for which U is convex 

with 0 1 and Rez 0 z U. 

Making use of the principle of subordination 

between analytic functions, let the subclasses 

 

MS 

p( ; ), MK 

p( ; ), MC 

p( , ; , ) and 

 

MC ( , ; , ) 

p 

of the class for 

p 

_____________________ 


*Corresponding author, S.M. Madian; Email: samar_math@yahoo.com

46 M.K. Aouf et al 

0 , p and , S , be defined as 

follows: 

z 

 

 

1 zf 

 

MS p( ; ) f p : z z U 

, 

 

p f z 

 

 

1 

f p 

: 

 

 

p 

 

MK p ( ; ) 

, 

 

 

 

zf z 

1 

 

z z U 

 

 

 

f z 

 

 

 

f : ( ; ) 

p g MS p 

 

 

MCp 

( , ; , ) 1 zf z 

 

 

s. t. 

 

 

 

z z U 

 

p g z 

 

and 

 

 

f : ( ; )s. t. 

p g MS p 

 

 

 

 

MC ( , ; , ) 

1 

 

p zf z 

. 

 

z z U 

 

p g z 

 

 

 

 

 

From these defnitions, we can obtain some 

well-known subclasses of p by special choices 

of the functions and as well as special 

choices of and see ([6], [13] and [22]). 

Let 1 ,A 1 ,..., q ,A q and 1 ,B 1 ,..., s ,B s 

q,s be positive and real parameters such 

that 

s 

1 

j1 

q 

B j A j 0. 

j1 

The Wright generalized hypergeometric 

function [23] (see also [25] 

 

, A ,..., , A ; , B ,..., , B ; z 

 

 

q s 1 1 q q 1 1 s s 

 

q 

s i , Ai ; , ; , 

1, q i Bi 

z 

 

1, s 

is defined by 

 

, A ; , B ; z 

 

 

q s i i 1, q i i 1, s 

 

q 

 

i 1 

 

s 

k 0 

i1 

 

 

 

i 

kAi 

z 

k 

. zU. 

k! 

kB 

i 

i 

 

If A i 1i 1,...,q and B i 1i 1,...,s, 

we have the relationship: 

 

 

q 

s i ,1 ; ,1 ; 

1, q i z 

 

1, s 

F ,..., ; ,..., ; z , 

q s 1 q s 

1 

where qF s 1 ,..., q ; 1 

,..., s ;z is the generalized 

hypergeometric function see for details 20 

 

and 24 and 

s 

i1 

 

i 

i1 

 

q 

 

 

i 

 

 

. 

 

(1.4) 

Consider the following linear operator due to 

Dziok and Raina [10] (see also [3] and [11]): 

p,q,s i ,A i 1,q 

; i ,B i 1,s 

: p p , 

defined by the convolution 

 

p, q, s i , Ai ; , 

1, q i B 

i f z 

 

1, s 

 

A B z 

f z 

 

p, q, s i , i ; , ; , 

1, q i i 

 

1, s 

where 

 

 

, A ; , B ; z 

p, q, s 

 

i i 1, q i i 1, s 

defined by Bansal et al. [7] as follows: 

 

 

 

was 

 

p, q, s i, Ai ; 

1, i, Bi 

; z 

 

q 

1, s 

(1.5) 

p 

 

z 

q 

s i, Ai ; 

, ; 

1, i 

Bi 

z 

1, 

z U 

. 

 

q 

s 

We observe that, for a function fz of the 

form 1.1, we have 

 

p, q, s 

i, Ai ; , 

1, q i 

B 

i 

f z 

 

1, s 

z 

p 

 

 

 

k 

1 p 

 

k, p 1 1 1 

 

k 

, A , B a z , 

k 

(1.6) 

where is given by 1.4 and k,p 1 ,A 1 ,B 1 

is defined by 

1 A1 

k p... 

q 

Aq k p 

k, p1 A1 B1 

1 B1 

k p... s 

Bsk pk p! 

, , . 

Corresponding to the function 

 

 

, A ; , B ; z 

p, q, s 

 

i i 1, q i i 1, s 

defined by (1.5), we introduce a function

Inclusion Properties of p-Valent Meromorphic Functions 47 

 

 

 

 

, A ; , B ; z 

 

by 

, A ; , B ; z 

 

p, q, s i i 1, q i i 1, s 

 

 

p, q, s i i 1, q i i 1, s 

 

 

 

 

p, q, s i , Ai ; , ; 

1, i B 

q i z 

 

1, s 

1 

 

0. 

p 

z 1 

z 

 

 

p, q, s 

 

i i 1, q i i 1, s 

defined by (1.6), we define the linear operator 

 

 

, , , ; , : 

p q s 

i Ai 

1, q i 

B 

i 

 

1, s p 

as follows: 

 

 

p 

 

 

p, q, s i , Ai ; 

, ( ) 

1, i B 

i f z 

 

q 1, s 

 

 

p, q, s i , Ai ; , ; ( ) 

1, q i Bi 

z 

f z 

 

1, s 

s 

q 

k p i 

k p 

Bi i (1.8) 

z 

Analogous to , A ; 

, B 

 

k 

1 p 

 

p 

i1 i1 

q 

s 

i k 

p 

Ai i 

 

i1 i1 

f p ; 0;z U . 

For convenience, we write 

 

1, A1 ,..., q, A ; 

 

 

 

q 

p, q, s 1, A1 , B1 f z 

p, q, 

s 

 

f z. 

 

1, B1 ,..., s, 

Bs 

 

 

One can easily verify from 1.8 that 

 

1, , 

 

1 p, q, s 1 1 1 

zA A B f z 

 

 

, A , B f z 

 

1 p, q, s 1 1 1 

( pA ) 

 

1 1 p, q, 

s 

 

 

and 

 

 

 

1, A , B f z ( A 0) 

1 1 1 1 

a z 

 

 

 

1 

p, q, s 

1, 1, 1 

 

p, q, s 1, 1, 

1 

z A B f z A B f z 

 

 

p, q, s 1 1 1 

k 

k 

(1.9) 

( p) , A , B f z ( 0). (1.10) 

Specializing the parameters p, q, s, A ( i 1,..., q), 

B ( i 1,..., s) 

and in (1.8) we have: 

i 

(i) For A 1( i 1,..., q), 

B 1( i 1,..., s) 

and 

i 

p p, p , we have 

 

,1,1 f z M ( ) f ( z), 

 

p 

 

p, q, s 1 p, q, s 1 

 

where the operator M p,q,s 1 was introduced 

by Patel and Patil [21] and Mostafa [17]; 

(ii) For A i = 1,…,q), B i = 1(I = 1,…,s), 

q 2, s 1, 

n p n p, p 

1 

i 

i 

and 2 1 ( 0) , we have 

 

 

 

p,2,1 n p, ; f z Inp 

1, 

f ( z), 

where the operator I np1, was introduced by 

Aouf and Xu [5] which for p 1 reduces to 

I n, n 1, 0 , where the operator I n, 

was introduced by Yuan et al. [28]; 

(iii) For A 1( i 1,..., q), B 1( i 1,..., s), 

i 

n p n p,p N, q 2, s 1 and 

n 

p 

we have [1,1;1] ( ) 

,2,1 

f z 

D 

1 2 1 

1, 

np1 

f ( z), 

where the operator 

i 

p 

n p 1 

D 

introduced by Yang [26] and Aouf ([1] and [2]); 

was 

 

(iv) For p 1, we have 1, qs , 

1, A1 , B1 

f z 

 

, qs , 

( 

1, A1 , B1 

) f z, 

where the operator 

( , A, B) 

was introduced by Aouf et al. 

, qs , 1 1 1 

[4]; 

(v) For A i 1 i 1,...,q,B i 1 i 1,...,s 

 

and p 1 , we have 

 

 

,1,1 

1, qs , 1 

f z 

H, qs ,( ) f z , where the operator H ,q,s 

was introduced by Cho and Kim [9], Muhamad 

[18] and Noor and Muhamad [19]. 

Also, we note that: 

(i) For 1 , then the operator 

 

1 

p, q, s 1 1 1 

 

, A, 

B reduces to the operator 

p,q,s 1 ,A 1 ,B 1 , defined by: 

 

, A , B f ( z) 

z 

p, q, s 1 1 1 

s 

 

p 

k 

p 

B 

 

i i i 

i1 i1 

q 

s 

k 

1 p 

i i i 

i1 i1 

 

 

k 

p 

A 

q 

az 

(ii) For A i 1 i 1,...,q,B i 1 i 1,...,s 

1 

and 1 , then the operator 

 

reduces to the operator 

 

N p,q,s 1 fz z p 

k1p 

k 

k 

, 

, 

p, q, s 1 

p q s 

, , 1 ,1,1 

N defined by: 

1 kp ... s kp 

1 kp ... q kp 

a k z k


 

 

0 

( Z {0, 1, 2,...}; 

i 

i 1,2,..., q ; p ). 

 

 

Next, by using the operator , A, 

B 

, 

p, q, s 1 1 1 

we introduce the following classes of 

meromorphic functions for 0 , p, 0 

In this paper we investigate several properties 

of the classes MS p,q,s 

 

 

1 ,A 1 ,B 1 ,;, MK p,q,s 1 , 

 

A 1 ,B 1 ;;, MC p,q,s 1 ,A 1 ,B 1 ;,;, and 

MC p,q,s 1 ,A 1 ,B 1 ;,;, associated with 

 

, A, B . Some applications 

the operator 

p, q, s 1 1 1 

involving integral operator are also considered. 

and , S : 

 

MS p, q, s 1 A1 B1 

 

 

 

( , , ; ; ) 

 

 

p p, q, s 1 1 1 

p 

f : , A , B f MS ( ; ) , 

MK 

 

 

p, q, s 1 A1 B1 

( , , ; ; ) 

 

 

p p, q, s 1 1 1 

p 

f : , A , B f MK ( ; ) , 

MC 

 

 

p, q, s 1 A1 B1 

( , , ; , ; , ) 

 

 

p p, q, s 1 1 1 

p 

f : , A , B f MC ( , ; , ) 

and 

MC 

 

 

p, q, s 1 A1 B1 

( , , ; , ; , ) 

 

 

 

p p, q, s 1 1 1 

p 

f : , A , B f MC ( , ; , ) . 

We can easily see that: 

f ( z) MK ( , A , B ; ; ) 

zf ( z) 

MS 

p 

and 

 

p, q, s 1 1 1 

 

 

p, q, s 1 1 1 

 

 

 

( , A , B ; ; ) 

(1.11) 

 

p, q, s 1 1 1 

f ( z) MC ( , A , B ; , ; , ) 

zf ( z) 

MC 

p 

MS 

( , A , B ; , ; , ). 

(1.12) 

 

p, q, s 1 1 1 

In particular, for 1 B A 1, 

we set 

1 

Az 

( , , ; ; ) 

1 Bz 

 

p, q, s 1 A1 B1 

 

p, q, s 1 1 1 

MS ( , A , B ; ; A, B) 

and 

MK 

1 

Az 

( , , ; ; ) 

1 

Bz 

 

p, q, s 1 A1 B1 

 

p, q, s 1 1 1 

MK ( , A , B ; ; A, B). 

 

 

 

 

2. INCLUSION PROPERTIES INVOLVING 

 

 

THE OPERATOR 

, A, 

B 

p, q, s 1 1 1 

In order to prove our results, we need the 

following lemmas. 

Lemma 1 [12]. Let be convex univalent in U 

with 0 1 and Relz 0 l, . 

If q is analytic in U with q0 1, then 

qz 

implies 

qz z . 

zq z 

lqz z , 

Lemma 2 [15]. Let be convex univalent in U 

and be analytic in U with Rez 0. 

If q is analytic in U and q0 0, then 

qz zzq z z , 

implies 

qz z . 

Theorem 1. Let S with 

1 

 

p 

p 

A1 

maxRe ( ) 

 

z min , 

zU 

p 

p 

 

 

 

 

1 

( , 0,0 p). 

A1 

Then 

MS 

1 

p, q, s 1 A1 B1 

( , , ; ; ) 

 

p, q, s 1 A1 B1 

MS ( , , ; ; ) 

 

p, q, s 1 A1 B1 

MS ( 1, , ; ; ).


Proof. To prove the first part, let 

1 

f MS ( , A , B ; ; ) 

and set 

p, q, s 1 1 1 

 

 

 

 

1 z p, q, s 

1, A1 , B1 

f ( z) 

 

q( z) 

( z U), 

p 

A B f z 

 

 

 

 

 

 

p, q, s 

1, 1, 1 

( ) 

 

(2.1) 

where q is analytic in U with q (0) 1. 

Applying (1.10) in (2.1), we obtain 

 

 

 

 

 

1 

1 z p, q, s 

1, A1 , B1 

f ( z) 

 

p 

A B f z 

 

 

 

 

 

 

1 

 

p, q, s 

1, 1, 1 

( ) 

 

zq( z) 

q( z) ( z U 

). 

( p ) q( z) 

p 

Since 

z 

p 

maxRe ( ) , we see that 

zU 

p 

(2.2) 

Rep qz p 0 z U. 

Applying Lemma 1 to (2.2), it follows that 

 

q( z) ( z) 

, that is f ( z) MS 

p, q, s( 1, A1 , B1 

, ; ) 

. Moreover, by using the arguments similar to 

those detailed above with (1.9), we can prove the 

second part. Therefore the proof is completed. 


 

maxRe ( z) 

zU 

 

1 

 

p 

1 1 

min , A p 

 

 

( , 0,0 p). 

p 

p 

 

A1 

 

 

Then 

MK 

MK 

1 

p, q, s 1 A1 B1 

( , , ; ; ) 

 

p, q, s 1 A1 B1 

( , , ; ; ) 

 

p, q, s 1 A1 B1 

MK ( 1, , ; ; ). 

Proof. Applying (1.11) and using Theorem 1, we 

observe that 

1 

p, q, s 1 1 1 

f ( z) MK ( , A , B ; ; ) 

zf () 

z 1 

MS p, q, s ( 1, A1 , B1 

; ; ) 

p 

zf () 

z 

MS p, q, s ( 1, A1 , B1 

; ; ) 

p 

 

p, q, s 1 1 1 

f ( z) MK ( , A , B ; ; ). 

Also 

 

p, q, s 1 1 1 

f ( z) MK ( , A , B ; ; ) 

zf () 

z 

MS p, q, s ( 1, A1 , B1 

; ; ) 

p 

zf () 

z 

MS p, q, s ( 1 1, A1 , B1 

; ; ) 

p 

 

p, q, s 1 1 1 

f ( z) MK ( 1, A , B ; ; ), 

which evidently proves Theorem 2. 

Taking 

1 

Az 

(z) 1 B A 1 , 

1 

Bz 

in Theorem 1 and Theorem 2, we have 

Corollary 1. Let 

 

1 

1 

p 1 

1 

min , A p 

A 

 

 

 

 

B 

 

p p 

1 

( , A 

0,0 

p, 1 B A 1). 

1 

Then 

1 

p, q, s 1 1 1 

MS ( , A , B ; ; A, B) 

 

p, q, s 1 1 1 

MS ( , A , B ; ; A, B) 

 

p, q, s 1 1 1 

MS ( 1, A , B ; ; A, B), 

and 

1 

p, q, s 1 1 1 

MK ( , A , B ; ; A, B) 

 

p, q, s 1 1 1 

MK ( , A , B ; ; A, B) 

 

p, q, s 1 1 1 

MK ( 1, A , B ; ; A, B). 

Next, by using Lemma 2, we obtain the following 

 

inclusion relations for the class MC p,q,s 1 ,A 1 , 

B 1 ;,;,. 

Theorem 3. Let , S with maxRe{ ( z)} 

 

 

1 

 

1 

p 

p 

min , A p 

MC 

p 

1 

p, q, s 1 A1 B1 

 

( , , ; , ; , ) 

 

p, q, s 1 A1 B1 

MC ( , , ; , ; , ) 

 

p, q, s 1 A1 B1 

MC ( 1, , ; , ; , ). 

zU 

1 

( , A 

0, 0 , p). 

1 

Then


Proof. To prove the first inclusion, let 

fz MC 1 p,q,s 1 , A 1 ,B 1 ;,;,. Then, 

1 

from the definition of MC ( , A , B ; , ; , ), 

p, q, s 1 1 1 

there exists a function gz MS 1 p,q,s 1 ,A 1 ,B 1 ;; 

such that 

1 

p 

Let 

z p,q,s 

1 1 ,A 1 ,B 1 fz 

1 p,q,s 1 ,A 1 ,B 1 gz 

 

 

 

 

 

z( , A , B f ( z)) 

 

q z z U 

 

z. 

 

 

1 

p, q, s 1 1 1 

( ) 

( 

 

p 

p, q, s 

1, A1 , B1 

g( z) 

 

), 

(2.3) 

where qz is analytic function in U with 

q(0) 1 . Using (1.10), we have 

 

 

p, q, s 1 1 1 

 

 

 

p, q, s 1 1 1 

[ ( p ) q( z) ] , A , B g( z) 

( + p) , A , B f ( z) 

 

, , ( ). (2.4) 

1 

p, q, s 1 

A1 B1 

f z 

Differentiating (2.4) with respect to 

multiplying by z , we obtain 

 

 

p, q, s 1 1 1 

( p ) zq( z) , A , B g( z) 

 

 

 

 

p, q, s 1 1 1 

[ ( p ) q( z) ] z( , A , B g( z)) 

 

 

1 

p, q, s 1 1 1 

z( , A , B f ( z)) 

 

 

 

 

p, q, s 1 1 1 

( + p) z( , A , B f ( z)) . 

Since 

1 1 1 

 

 

 

z and 

g( z) MS ( , A , B ; ; ) MS 

1 

 

p, q, s 1 1 1 p, q, 

s 

( , A, B; ; ), by Theorem 1, we set 

 

 

 

 

 

 

1 

p, q, s 1 1 1 

( z) 

, 

 

p 

p, q, s 

1, A1 , B1 

g( z) 

 

 

 

z( , A , B g( z)) 

 

 

(2.5) 

where ( z) ( z) 

in U with the assumption 

S . Then, by using (2.3), (2.4) and (2.5), we 

have 

 

 

 

1 

1 

 

z( p, q, s 

1, A1 , B1 

f ( z)) 

 

 

 

1 

p 

p, q, s 

1, A1 , B1 

g( z) 

 

 

 

zq ( z) 

q( z) ( z). 

( p ) ( z) 

p 

 

 

(2.6) 

Since 0 and ( z) ( z) 

in U 

p 

with maxRe{ ( z)} , then 

zU 

p 

Rep z p 0 z U. 

Hence, by taking 

z 1 

p z p , 

in (2.6) and applying Lemma 2, we have 

q( z) ( z) 

in U , so that 

 

f ( z) MC ( , A , B ; , ; , ). The second 

p, q, s 1 1 1 

inclusion can be proved by using arguments 

similar to those detailed above with (1.9). This 

compelets the proof of Theorem 3. 

Theorem 4. Let , S with 

 

1 

1 

p 

maxRe{ ( )} min , A p 

z 

zU 

1 

( , A 

0, 0 , p). 

1 

Then 

MC 

1 

p, q, s 1 A1 B1 

( , , ; , ; , ) 

 

p, q, s 1 A1 B1 

 

p, q, s 1 A1 B1 

p 

MC ( , , ; , ; , ) 

MC ( 1, , ; , ; , ). 

 

p 

Proof. Just as we derived Theorem 2 as a 

consequence of Theorem 1 by using the 

equivalence (1.11), we can also prove Theorem 4 

by using Theorem 3 in conjunction with the 

equivalence (1.12). 

3. PROPERTIES FOR THE INTEGRAL 

OPERATOR 

F , p 

Let F , p 

be the integral operator defined by (see 

[14] and [27]): 

z 

p1 

, 

p 

( )( ) 

( ) 

 

p 

z 

 

0 

F f z t f t dt 

 

p k 

( z z ) f ( z) 

k 

p 

k 

1 p 

 

(3.1)


f p ; 0;z U . 

From (3.1), we observe that 

 

 

 

p 

f z 

 

 

p, q, s 1 1 1 , 

z( , A , B F ( f )( z)) 

 

p, q, s 1 A1 B1 

, , ( ) 

 

p, q, s 1 1 1 , 

p 

( p) , A , B F ( f )( z) 0 . 

The proof of Theorem 5 below, is much akin 

to that of Theorem 1, so, we omit it. 


p 

maxRe{ ( z)} 

( 0, 0 

p). 

If 

zU 

p 

f ( z) MS ( , A , B ; ; ), 

then 

 

p, q, s 1 1 1 

F ( f )( z) MS ( , A , B ; ; ). 

 

, p 

 

p, q, s 

1 1 1 

 

Next, we derive an inclusion property 

involving F 

, which is obtained by applying 

, p 

(1.11) and Theorem 1. 

Theorem 6. Let S with maxRe{ ( z )} 

zU 

p 

p 

0, 0 p. 

 

If f ( z) MK 

p, q, s( 1, A1 , B1 

; ; ), 

 

then F 

, p 

f z MK 

p, q, s 1 

A1 B1 

 

( )( ) ( , , ; ; ). 

1 

Taking ( z) A 

1B 

( 1 B A 1) and from 

Theorems 5 and 6, we have 


1 

A 

1B 

( 0, 0 

p 

p 

p, 1 B A 1). 

Then if f z MS 

 

, , 

( , A , B ; ; A, B) 

(or 

then 

1 1 1 

 

p, q, s 1 1 1 

 

, p 

p, q, s 

1 1 1 

 

() 

p q s 

MK ( , A , B ; ; A, B)) 

, 

F ( f )( z) MS ( , A , B ; ; A, B) 

(or 

MK ( , A , B ; ; A, B)). 

 

p, q, s 1 1 1 

Finally, we obtain Theorems 7 and 8 below by 

using the same techniques as in the proof of 

Theorems 3 and 4. 

Theorem 7. Let , S with 

p 

maxRe{ ( z)} 

 

zU 

p 

0, 0 , p. If f z MC 

, , 

() p q s 

 

1 A1 B1 

then 

, p 

p, q, 

s 

( , , ; , ; , ), 

( , A, B; , ; , ). 

1 1 1 

F ( f )( z) 

MC 

Theorem 8. Let , S with maxRe{ ( z)} 

p 

 

p 

MC 

zU 

f z 

 

0, 0 , p. If () 

 

p, q, s ( 1 , A1 , B1 

; , ; , ), 

1 1 1 

then F , p ( f )( z) 

MC 

, , 

( , A, B; , ; , ). 

p q s 

Remark 1. (i) If we take p 1, A n 1 

n 1,...,q and B n 1 n 1,...,s in 

the above results of this paper, we obtain the 

results obtained by Cho and Kim [9]; 

(ii) If we take p 1 , 1( 1,..., ), 

Ai 

i q Bi 

1( i 1,..., s), q 2, s 1, n 1( n 1) and 

2 1 0 in the above results of 

this paper, we obtain the results obtained by Yuan 

et al. [28]; 

(iii) If we take p 1 in the above results of this 

paper, we obtain the results obtained by Aouf et 

al. [4]. 

Remark 2. Specializing the parameters 

p, q, s, Ai 

( i 1,..., q), Bi 

( i 1,..., s) 

and in 

the above results of this paper, we obtain the 

results for the corresponding operators 

 

n p 1 

M ( ), I 

 

and 

which are 

p, q, s 1 n p 1, 

defined in the introduction. 

4. CONCLUSIONS 

1 

D 

In this paper, using the Wright generalized 

hypergeometric function we define a new operator 

which contains many other operators as special 

cases of it. Also, we define some classes of 

meromorphic functions associated to this operator 

by using the principle of subordination and 

investigate several inclusion properties of these 

classes. Some applications involving integral 

operator are also considered. Our results 

generalize many previous results. 

5. ACKNOWLEDGEMENTS 

The authors would like to thank the referees of the 

paper for their helpful suggestions.


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Reade. On a Briot-Bouquet differential 

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ISSN: 0377 - 2969 



Some Inclusion Properties of Certain Operators 

M.K. Aouf 1 , R.M. El-Ashwah 1 and E.E. Ali 1,2* 

1 Mathematics Department, Faculty of Science, Mansoura University, Mansoura 33516, Egypt 

2 Mathematics Department, Faculty of Science, University of Hail, Kingdome of Saudi Arabia 

Abstract: In this paper we introduce several new subclasses of analytic p vhalent functions which are 

defined by means of a general integral operators I ( a, 

b, 

c) 

( a, 

b, 

c 

\ Z 

 

, p, 

p ) 

and 

, 

p 

0 

investigate various inclusion properties of these subclasses. Many interesting applications involving these 

and other families of p valent operators are also considered. 

Keywords: Analytic function, starlike of order , 

convex of order , subordinate, Hadamard product, 

integral operator. 

2000 Mathematics Subject Classification : 30C45 


Let A( p) 

denote the class of functions of the form: 

f ( z) 

z 

p 

 

a 

k1 

k 

p 

z 

k 

p 

( p {1,2,...}), 

(1.1) 

which are analytic and p valent in the open unit 

disc U { z :| z | 1} 

. A function f ( z) 

A( 

p) 

is 

said to be in the class S 

( ) of p valently 

starlike of order , if it satisfies 

zf ( 

z) 

 

Re 

 

f ( z) 

 

 

We write 

p 

(0 p; 

z U). 

(1.2) 

S ( 0) 

S , the class of p valently 

 

p 

 

p 

starlike in U . A function f ( z) 

A( 

p) 

is said to 

be in the class K p( ) of p valently convex of 

order , if it satisfies 

It follows form (1.2) and (1.3) that 

zf ( 

z) 

 

f ( z) 

K 

p( 

) S 

p( 

) 

(0 

p) . (1.4) 

p 

The classes S 

p( 

) and K p( ) were studied by 

Owa [1] and Patil and Thakare [2]. 

Furthermore, a function f ( z) 

A( 

p) 

is said 

to be p valently close-to-convex functions of 

order and type in U , if there exists a 

 

function g ( z) 

S ( ) such that 

zf ( 

z) 

 

Re 

 

g( 

z) 

 

 

p 

(0 , 

p; 

z U) . (1.5) 

We denote by B ( , 

) , the subclass of A ( p) 

p 

consisting of all such functions. The class 

B ( , 

) was studied by Aouf [3]. 

p 

zf ( z) 

 

Re1 

 

f ( 

z) 

 

 

(0 p; 

z U). 

(1.3) 

―――――――――――――――― 

Received, January 2012; Accepted, March 2012 

*Corresponding author, E.E. Ali; E-mail: ekram_008eg@yahoo.com 

Suppose that f (z) 

and g (z) 

are analytic in 

U . Then we say that the function g (z) 

is 

subordinate to f (z) 

if there exists an analytic


function w(z) 

in U with w( z) 

z for all 

z U , such that ( z) 

f ( w( 

z)) 

g , denoted 

g f of g( z) 

f ( z) 

. In case f (z) 

is univalent 

in U we have that the subordination g( z) 

f ( z) 

is equivalent to g( 0) f (0) 

and g( U) 

f ( U) 

(see [4]; see also [5],[6, p. 4]). 

For the functions f j 

( z) 

( j 1,2) 

defined by 

 

p 

k 

p 

f 

j 

( z) 

z ak 

p, 

jz 

( p) 

(1.6) 

k1 

we denote the Hadamard product (or convolution) 

of f 1( 

z ) and f ( z) 

by 2 

 

p 

k 

p 

f f )( z) 

z a a z . (1.7) 

( 

1 2 

k 

p,1 

k 

p, 

2 

k1 

Let M be the class of analytic functions 

(z) in U normalized by ( 0) 1, and let S be 

the subclass of M consisting of those functions 

(z) which are univalent in U and for which 

(U) is convex and Re ( z) 

0 ( zU) 

. 

Making use of the principle of subordination 

between analytic functions, we introduce the 

subclasses S ( ), 

K ( ) 

and C ( , 

) of the 

p 

p 

class A ( p) 

for , S , which are defined by 

 

zf ( 

z) 

 

S p 

( ) 

f : f A( 

p) 

and ( 

z) 

in U , 

 

pf ( z) 

 

 

1 zf ( z) 

 

 

K p 

( ) 

f : f A( 

p) 

and 1 

( 

z) 

in U , 

 

p f ( 

z) 

 

 

f : f A( p) and h 

 

 

Cp 

( , ) f() 

z 

. 

 

K p ( ) s. t. ( z) in U 

h() 

z 

 

 

 

p 

1 

z 

Kp 

Kp 

, 

1 

z 

p ( p 2 ) 

z 

K p 

K p( ) (0 p), 

1 

z 

1z 

1z Cp 

, Cp 

, 

1z 

1z 

p ( p 2 ) z p ( p 2 ) 

z 

Cp 

, Cp( , ) (0 , p). 

1z 

1z 

 

 

Furthermore, for the function classes S p 

[ A, 

B, 

] 

and K p 

[ A, 

B, 

] 

investigated by Aouf ([9, 10], it is 

easily seen that 

1[ 

B( 

AB)(1 

)] 

 

p 

S 

p 

1 

Bz S 

p[ 

A, 

B, 

] ( 1 

B A 1;0 

p) 

 

(see Aouf [9]), 

And 

1[ 

B( 

AB)(1 

)] 

p 

K 

p 

1 

Bz K 

p[ 

A, 

B, 

] ( 1 

B A 1;0 

p) 

 

(see Aouf [10]). 

For real or complex number a , b, 

c other than 

0, 

1, 2,... 

, the hypergeometric series is defined 

by 

( a) 

k 

( b) 

k k 

2 

F1 

( a, 

b; 

c; 

z) 

 

z , 

(1.8) 

k0 

( c) 

(1) 

k 

where ( x) 

k 

is Pochhammer symbol defined by 

k 

( 

x k) 

x( 

x 1)...( 

x k 1) 

( k N; 

xC), 

( x) 

k 

 

( 

x) 

1 

( k 0; k C 

\{0}). 

We note that the series (1.8) converges 

We note that for p 1, the classes 

S 

 

1 

( ) 

S 

 

( ), 

K1( 

) 

K( 

) 

and 

C1 ( , 

C( 

, 

) are investigated by Ma and 

Minda [7] and Kim et al [8]. 

Obviously, for special choices for the 

functions and involved in the above 

definitions, we have the following relationships: 

1 

z 

Sp 

Sp 

, 

1 

z 

p ( p 2 ) 

z 

S p 

S p( ) (0 p), 

1 

z 

absolutely for all z U so that it represents an 

analytic function in U (see, for details, [11, 

Chapter 14]). 

Now we set 

p 

z 

f , p( 

z) 

 

p 

(1 z) 

( p) 

(1.9) 

and define f , p( 

z) 

by means of the Hadamard 

product 

( 1) 

p 

f , p( z) 

f, 

p 

( z) 

z 

2F1 

( a, 

b; 

c; 

z) 

( zU) 

, (1.10) 

This leads us to a family of linear operators

Inclusion Properties of Certain Operators 55 

( 1) 

, p 

, 

p 

I ( a, b, c) f ( z) f ( z) 

 

0 

( a, b, c R \ Z , p, p , z U). 

(1.11) 

After some computations, we obtain 

 

p ( a) 

k 

( b) 

k 

k 

p 

I , p( 

a, 

b, 

c) 

f ( z) 

z 

ak 

pz 

. (1.12) 

k1 

( c) 

( p) 

From (1.12), we deduce that 

( a, 

p, 

a) 

f ( z) 

f ( z) 

( p, 

p 

I 

, p 

 

and 

zf ( 

z) 

I1, 

p 

( p 1, p 1, 

p) 

f ( z) 

, 

p 

z( I ( a, b, c) f ( z)) ( p) I ( a, b, c) f ( z) 

1, p 

, 

p 

I ( a, b, c) f ( z) ( p), 

1, 

p 

and 

z( I ( a, b, c) f ( z)) aI ( a 1, b, c) f ( z) 

, p 

, 

p 

( a p) I ( a, b, c) f ( z). 

, 

p 

We note that; 

k 

k 

) 

(1.13) 

(1.14) 

(i) I a, 

p 1, 

a) 

f ( z) 

I ( n ) , where 

n, p( n 

p1 

p 

I 

n p1 

is the Noor integral operator of 

( n p 1) 

th 

order (see Liu and Noor [12] 

and Patel and Cho [13]); 

n 

p1 

(ii) I ( p 1, 

n p,1) 

f ( z) 

D f ( z) 

( n ) , 

1, 

p 

p 

1 

where D n 

p 

f ( z) 

is the ( n p 1) 

th 

order Ruscheweyh derivative of a function 

f ( z) 

A( 

p) 

(see Kumar and Shukla [14]); 

(iii) I a,2, 

a) 

f ( z) 

I f ( z) 

( n 1) 

( n , 1 n 

 

is the Noor integral operator of 

(see [15]); 

( , 

p) 

(iv) I ( a, 

p 1, 

a) 

f ( z) 

f ( ) 

1 

, 

p 

z 

z 

( k p 1) ( p 1 ) 

z ak 

pz 

( p 1) ( k p 1 ) 

k1 

z F (1, p 1; p 1 ; z) f ( z) 

2 1 

( p 1; z U 

). 

, where I 

n 

n th order 

p k p 

p 

( , 

p) 

The operator 

z 

was introduced and 

studied by Patel and Mishra [16]: 

(v) 

I, 

p ( p, p, p 1) f ( z) 

, 

J f ( z) ( p) 

, 

p 

where J , p 

is the generalized Bernardi- 

Libera-Livingston operator defined by (3.1) 

(see [17]); 

I,1 ( , b, b) f ( z) I, 

f ( z) 

(vi) 

, 

( 1, 0, f ( z) A(1) A) 

where I 

, 

is the Choi-Saigo-Srivastava 

operator (see [17]). 

We also note that: 

p 

I , p( , 

b, 

b) 

f ( z) 

I, 

 

f ( z) 

( p, 

0, f ( z) 

A( 

p)) 

, 

p 

where I , 

is the generalized Choi-Saigo- 

Srivastava operator (see [17]) defined by 

 

p 

p ( ) 

k 

k 

p 

I, 

f ( z) 

z ak 

pz 

( p; 

0; z U) . 

k1 

( p) 

k 

Next, by using the general operator 

( a, 

b, 

) , we introduce the following classes of 

I , p 

c 

analytic 

S 

 

, 

p 

p valent functions for 

f : f A( p) and 

( a, b, c; ) 

, 

 

I, 

p( a, b, c) f ( z) S 

p( ) 

 


K, 

p ( a, b, c; ) 

, 

I, 

p( a, b, c) f ( z) K 

p( ) 

 

 

 

And 

C 

, 

p 


( a, b, c; , ) 

. 

I, 

p( a, b, c) f ( z) Cp( , ) 

 

 

 

We also note that 

zf ( 

z) 

 

f ( z) 

K, p( 

a, 

b, 

c; 

) 

S, 

p( 

a, 

b, 

c; 

). 

(1.15) 

p 

In particular, we set 

1 

z 

Sn, pa, p 1, 

a; 

Sn 

p1 

( n p), 

1 

z 

S 

1 

Az 

a 

b, 

c; 

S 

1 

Bz 

a, 

b, 

c; 

A, 

B ( 1 

B A 1), 

 

 

, 

p 

, 

, 

p


and 

1 

Az 

K 

pa 

b, 

c; 

K 

1 

Bz 

a, 

b, 

c; 

A, 

B ( 1 

B A 1). 

, 

, 

, 

p 

 

Inclusion properties was investigated by 

several authors (e.g. see [18], [19], [20] and [21]). 

In this paper, we investigate several inclusion 

 

properties of the classes ( a, 

b, 

c; 

), 

S, p 

 

K p 

( a, 

b, 

c; 

) and C p 

( a, 

b, 

c; 

, 

) associated 

, 

, 

with the general integral operator ( a, 

b, 

) 

I , p 

c . 

Some applications involving these and other 

families of integral operators also considered. 

2 . INCLUSION PROPERTIES INVOLVING 

I , 

p 

To establish our main results, we shall need the 

following lemmas. 

Lemma 1 [22]. Let h be convex univalent in 

U with h ( 0) 1 

and 

Re 

h( 

z) 

0 ( , 

C) 

. 

If q (z) 

is analytic in U with q ( 0) 1, then 

zq( 

z) 

q( z) 

h( 

z) 

( z U) 

q( 

z) 

 

implies that q( z) 

h( 

z) 

( z U) 

. 

Lemma 2 [23]. Let h be convex in U with 

h ( 0) 1. Suppose also that Q (z) 

is analytic in U 

with ReQ 

( z) 

0 ( zU) 

. If q (z) 

is analytic in 

U with q ( 0) 1, then 

q( z) 

Q( 

z) 

zq( 

z) 

h( 

z) 

( zU) 

implies that q( z) 

h( 

z) 

( z U) 

. 

Theorem 1. Let p, 

a p and p 

. Then 

 

 

, p 

, 

p 

S ( a 1, b, c; ) S ( a, b, c; ) 

 

1, 

p 

S ( a, b, c; ) ( S). 

Proof. First of all, we show that 

 

, 

p 

S ( a 1, b, c; ) 

 

, 

p 

S ( a, b, c; ) ( S; p; a p; p N). 

Let f ( z) 

S 

 

, p( 

a 1, 

b, 

c; 

) 

and set 

 

z I 

pI 

, 

p 

, 

p 

 

( a, 

b, 

c) 

f ( z) 

 

 

( a, 

b, 

c) 

f ( z) 

q( 

z) 

, 

(2.1) 

2 

where q ( z) 

1 

q1z 

q2z 

... 

is analytic in U 

and q ( z) 

0 for all z U . Using the identity 

(1.14) in (2.1), we obtain 

I, 

p( 

a 1, 

b, 

c) 

f ( z) 

a 

pq( 

z) 

a p . 

I ( a, 

b, 

c) 

f ( z) 

, 

p 

(2.2) 

Differentiating (2.2) logarithmically with 

respect to z , we have 

 

 

I 

( a 1, 

b, 

c) 

f ( z) 

zI 

( a, 

b, 

c) 

f ( z) 

 

z 

I 

, 

p 

, 

p 

( a 1, 

b, 

c) 

f ( z) 

Since 

 

I 

, 

p 

, 

p 

( a, 

b, 

c) 

f ( z) 

zq( 

z) 

q( 

z) 

 

. 

pq( 

z) 

a p 

zq( 

z) 

 

pq( 

z) 

a p 

(2.3) 

a p, ( z) 

S 

, and f ( z) 

S 

, p( 

a 1, 

b, 

c; 

) 

, 

from (2.3) we see that 

Re 

p 

( z) 

a p 0 ( zU) 

and 

zq( 

z) 

q( z) 

 

( 

z) 

( z U) 

pq( 

z) 

a p 

Thus, by using Lemma 1 and (2.1), we 

observe that 

q( z) 

( 

z) 

( zU) , 

so that 

 

f ( z) 

S 

, 

( a, 

b, 

c; 

) . 

p 

This implies that 

 

 

S, p( a 1, b, 

c; 

) 

S, 

p( 

a, 

b, 

c; 

) . 

To prove the second part, let 

 

f ( z) S ( a, b, c; ) ( p; a p; p ) and 

put 

 

z I 

pI 

1, 

p 

1, 

p 

, 

p 

( a, 

b, 

c) 

f ( z) 

 

 

( a, 

b, 

c) 

f ( z) 

g( 

z) 

, 

2 

where g ( z) 

1 

d z d z ... is analytic in U 

1 2


and g ( z) 

0 for all z U . Then, by using 

arguments similar to those detailed above with the 

identity (1.13), it follows that 

g( z) 

( 

z) 

( zU) , 

 

which implies that f z) 

S ( a, 

b, 

c; 

) . Hence 

we conclude that 

( 

1, 

p 

 

 

 

 

S, p( a 1, 

b, 

c; 

) 

S, 

p( 

a, 

b, 

c; 

) 

S 

1, 

p( 

a, 

b, 

c; 

) 

, 

which completes the proof of Theorem 1. 

1z Putting n, c a, 

b p 1 

and ( z) 

 

1 z 

( zU) 

in 

Theorem 1, we obtain the following corollary. 


S 

S 

 

 

n p1 

 

n 

p 

. 

n p 

and p 

. Then 

Remark 1. Putting p 1 

in Corollary 1, we 

obtain the result obtained by Noor [15]. 


a p and p 

. Then 

C ( a 1, b, c; ) C ( a, b, c; ) 

, p 

, 

p 

C ( a, b, c; ) ( S). 

1, 

p 

Proof. Applying (1.15) and Theorem 1, we 

observe that 

f ( z) C , 

p 

( a 1, b, c; ) 

I 

, 

p 

( a 1, b, c) f ( z) K p( ) 

z ( I 

, 

( a 1, b , c ) f ( z )) S 

 

p p 

p( ) 

zf 

() 

z 

I 

, p 

( a 1, b, c) p S 

 

 

p( ) 

 

zf () 

z 

p S 

 

, 

p 

( a 1, b, c; ) 

zf () 

z 

p S 

 

, 

p 

( a, b, c; ) 

zf 

() 

z 

I 

, 

( a, b, c) S 

 

p p p ( ) 

 

 

z 

p I 

, p 

( a , b , c ) f ( z ) S 

 

 

p( ) 

I 

, 

p 

( a, b, c) f ( z) K p( ) 

f ( z) C , 

p 

( a, b, c; ) 

and 

f ( z) K ( a, b, c; ) 

 

1, 

p 

, 

p 

zf () 

z 

p 

zf () 

z 

p 

1, 

p 

 

, 

p 

S ( a, b, c; ) 

 

1, 

p 

S ( a, b, c; ) 

 

z 

 

I ( a, b, c) f ( z) S ( ) 

p 

I ( a, b, c) f ( z) K ( ) 

f ( z) K ( a, b, c; ), 

1, 

p 

which evidently proves Theorem 2. 

Taking 

1 

Az 

( z) 

( 1 

B A 1; 

z U) 

1 

Bz 

in Theorem 1 and 2, we have 

 

p 

Corollary 2. Let p , a p, 

p 

and 

1 

B A 1. 

Then 

 

 

1, , ; , p , , ; , 

a, b, c; A, 

B 

 

, p 

, 

S a b c A B S a b c A B 

S 

 

1, 

p 

and 

K a 1, b, c; A, B K a, b, c; A, 

B 

p 

a, b, c; A, B. 

, p 

, 

K 

1, 

p 


a p and p 

. Then 

C ( a 1, b, c; , ) C ( a, b, c; , ) 

, p 

, 

p 

C ( a, b, c; , ) ( , S). 

1, 

p 

Proof. We begin by proving that 

C ( a 1, b, c; , ) C ( a, b, c; , ) 

, p 

, 

p 

( p; a p; p ; , S). 

Let f ( z) 

C 

, p( 

a 1, 

b, 

c; 

, 

) . Then, in view of 

 

 

(1.7), there exists a function h ( z) 

S ( ) 

such 

that 

z I 

( 

, 

p 

( a 1, 

b, 

c) 

f ( z)) 

 

( z) 

ph( 

z) 

p 

p 

( z U) . 

Choose the function g(z) 

such that


I , p 

( a 1, 

b, 

c) 

g( 

z) 

h( 

z) 

. Then g( z) 

S 

 

, p( 

a 1, 

b, 

c; 

) 

and 

z( 

I 

pI 

z( 

I 

pI 

, 

p 

, 

p 

( a 1, 

b, 

c) 

f ( z)) 

 

( z) 

( z U) . (2.4) 

( a 1, 

b, 

c) 

g( 

z) 

Now let 

 

z 

I 

 

 

, 

p 

, 

p 

( a 1, 

b, 

c) 

f ( z)) 

 

q( 

z), 

( a 1, 

b, 

c) 

g( 

z) 

(2.5) 

2 

where q ( z) 

1 

q1z 

a2z 

... 


U and q ( z) 

0 for all z U . Thus by using the 

identity (1.14), we have 

z( 

I 

pI 

, 

p 

, 

p 

( ) 

I 

, 

( a 1, b, 

c) 

zf 

p 

( a 1, b, 

c) 

f ( z)) 

p z 

 

 

 

 

 

 

 

( a 1, 

b, 

c) 

g( 

z) 

I ( a 1, 

b, 

c) 

g( 

z) 

, 

p 

 

( ) 

( ) 

, 

( , , ) 

zf 

p z 

 

 

z I a b c 

 

 

 

 

, 

( , , ) 

zf 

p z 

 

 

 

p 

I a b c 

 

p 

 

 

 

 

( a p) 

I, 

p 

( a, 

b, 

c) 

g( 

z) 

I, 

p 

( a, 

b, 

c) 

g( 

z) 

 

. 

z( 

I, 

p 

( a, 

b, 

c) 

g( 

z)) 

 

( a p) 

I, 

p 

( a, 

b, 

c) 

g( 

z) 

(2.6) 

 

 

Since g z) 

S 

( a 1, 

b, 

c; 

) 

S ( a, 

b, 

c; 

) 

( ) 

( 

, p 

, 

p 

S , 

by Theorem 1, we set 

z( 

I 

, 

( a, 

b, 

c) 

g( 

z)) 

 

p 

G( 

z) 

, 

pI ( a, 

b, 

c) 

g( 

z) 

, 

p 

, 

p 

( ) 

( a, 

b, 

c) 

zf 

p z 

 

 

( a p) 

I 

 

z 

( ) 

( a, 

b, 

c) 

zf 

p z 

 

 

 

I 

( a, 

b, 

c) 

g( 

z) 

( a p) 

I ( a, 

b, 

c) 

g( 

z) 

, 

p 

, 

p 

, 

p 

where G( z) 

( 

z) 

( z U) 

for S. 

Then, by 

virture of (2.5) and (2.6), we observe that 

zf ( 

z) 

 

I 

, 

( a, 

b, 

c) 

 

p 

q( 

z) 

I, 

p( 

a, 

b, 

c) 

g( 

z) 

(2.7) 

p 

and 

 

( ) 

, 

( , , ) 

zf 

p z 

 

 

 

 

 

zI 

a b c 

p 

 

 

 

z( 

I ( a 1, 

b, 

c) 

f ( z)) 

 

( a p) 

q( 

z) 

, p 

I, 

p 

( a, 

b, 

c) 

g( 

z) 

 

. (2.8) 

pI ( a 1, 

b, 

c) 

g( 

z) 

pG( 

z) 

a p 

, 

p 

Differentiating both sides of (2.7) with respect 

to z , we obtain 

( ) 

z I 

, 

( a, 

b, 

c) 

zf 

p p z 

 

 

 

 

I ( a, 

b, 

c) 

g( 

z) 

z( 

I 

pI 

, 

p 

pG( 

z) 

q( 

z) 

zq( 

z) 

. 

Making use of (2.4), (2.8) and (2.9), we get 

, 

p 

, 

p 

(2.9) 

( a 1, 

b, 

c) 

f ( z)) 

pG( 

z) 

q( 

z) 

zq( 

z) 

( a p) 

q( 

z) 

 

( a 1, 

b, 

c) 

g( 

z) 

pG( 

z) 

a p 

zq( 

z) 

q( z) 

 

( z) 

( z U) . (2.10) 

pG( 

z) 

a p 

Since 

a p, p 

and G( z) 

( 

z) 

( z U) 

, 

pG( 

z) 

a p 0 ( zU) 

. 

Re 

Hence, by taking 

1 

Q( 

z) 

 

pG( 

z) 

a p 

in (2.10), and applying Lemma 2, we can show 

that 

p( z) 

( z) 

( zU) , 

so that 

f ( z) 

C 

, p( 

a, 

b, 

c; 

, 

) ( , 

S) . 

 

For the second part, by using arguments 

similar to those detailed above with the identity 

(1.13), we obtain: 

C a, 

b, 

c; 

, 

) C 

( a, 

b, 

c; 

, 

) ( , 

) . 

, p( 1, 

p 

S 

The proof of Theorem 3 is thus completed. 

3. INCLUSION PROPERTIES INVOLVING 

J , p 

In this section, we consider the generalized 

Bernardi-Libera-Livingston integral operator 

J , p 

( p) 

defined by (see [24],[25],and [26]). 

p 

z 

1 

J , p( 

f )( z) 

t 

f ( t) 

dt ( f A( 

p); 

p) . (3.1) 

 

z 0 

Theorem 4. Let p, p, 

a p and 

p . If f ( z) 

S 

 

, p( 

a, 

b, 

c; 

) 

( S) 

, then


J 

, p( f )( z) 

S 

 

 

, 

p( 

a, 

b, 

c; 

) 

( S) . 

 

Proof . Let f ( z) 

S 

, 

( a, 

b, 

c; 

) 

for S 

, and 

set 

z( 

I 

pI 

, 

p 

, 

p 

( a, 

b, 

c) 

J 

( a, 

b, 

c) 

J 

, p 

, p 

p 

( f )( z)) 

 

q( 

z) 

, 

( f )( z) 

(3.2) 

2 

where q ( z) 

1 

q1z 

q2z 

... 


U and q ( z) 

0 for all z U . From (3.1), we 

obtain 

z( I, p ( a, b, c) J, p ( f )( z)) ( p) I, 

p ( a, b, c) f ( z) 

(3.3) 

I ( a, b, c) J ( f )( z) ( z U) . 

, p 

, 

p 

By applying (3.2) and (3.3), we obtain 

I, 

p( 

a, 

b, 

c) 

f ( z) 

( p) 

pq( 

z) 

 

. 

I ( a, 

b, 

c) 

J ( f )( z) 

, 

p 

, p 

(3.4) 

Differentiating (3.4) logarithmically with respect 

to z , we obtain 

z( 

I 

I 

, 

p 

, 

p 

( a, 

b, 

c) 

f ( z)) 

zq( 

z) 

q( 

z) 

. 

( a, 

b, 

c) 

f ( z) 

pq( 

z) 

 

Since 

from (3.5), we have 

Re 

(3.5) 

 

p, ( z) 

S 

, and f ( z) 

S 

, 

( ) 

, 

zq( 

z) 

pq( 

z) 

 

p 

p( 

z) 

0 and q( 

z) 

( 

z) 

( z U) 

. 

Hence, by virbure of Lemma 1, we conclude 

that q( z) 

( 

z) 

( zU) 

, 

which implies that 

J 

, p( f )( z) 

S 

 

 

, 

p( 

a, 

b, 

c; 

) 

( S) . 

Next, we derive an inclusion property 

involving , which is given by 

J , p 

Theorem 5. Let p, p, 

a p and 

p . If f ( z) 

K 

, p( 

a, 

b, 

c; 

) 

( S) 

, then 

J 

, p( f )( z) 

K, 

p( 

a, 

b, 

c; 

) 

( S) . 

 

Proof . By applying Theorem 4, it follows that 

zf ( 

z) 

 

f ( z) 

K, p( 

a, 

b, 

c; 

) 

S, 

p( 

a, 

b, 

c; 

) 

p 

zf ( z) 

 

J, p 

S, 

p( a, b, c; ) 

p 

z 

J , p( f )( z ) S 

 

, 

p( a , b , c ; ) 

p 

J f )( z) 

K 

( a, 

b, 

c; 

) 

( 

, p( , 

p 

S 

which proves Theorem 5. 

Finally, we prove 


) , 

p, p, 

a p and 

p . If f ( z) 

C 

, p( 

a, 

b, 

c; 

, 

) ( , 

S) 

, then 

 

J 

, p( f )( z) 

C 

, p( 

a, 

b, 

c; 

, 

) ( , 

S) . 

Proof. Let f ( z) 

C 

, 

( a, 

b, 

c; 

, 

) for , S 

. 

p 

Then, in view of (1.7), there exists a function 

 

g( z) 

S 

, 

( a, 

b, 

c; 

) 

such that 

z( 

I 

pI 

, 

p 

, 

p 

p 

( a, 

b, 

c) 

f ( z)) 

 

( z) 

( z U) . (3.6) 

( a, 

b, 

c) 

g( 

z) 

Thus we set 

z( 

I ( a, 

b, 

c) 

J 

pI 

, 

p 

, 

p 

( a, 

b, 

c) 

J 

, p 

, p 

( f )( z)) 

 

q( 

z) 

, 

( f )( z) 

2 

where q ( z) 

1 

q1z 

q2z 

... 


U and q ( z) 

0 for all z U . Applying (3.3), we 

get 

( ) 

I 

, 

( a, 

b, 

c) 

zf 

p 

z( 

I 

, 

( a, 

b, 

c) 

f ( z)) 

p z 

 

 

p 

 

 

pI ( a, 

b, 

c) 

g( 

z) 

I ( a, 

b, 

c) 

g( 

z) 

, 

p 

 

z 

I 

 

 

z 

 

 

z 

I 

 

, 

p 

( a, 

b, 

c) 

J 

, 

p 

( a, 

b, 

c) 

J 

 

I 

( a, 

b, 

c) 

J ( g)( 

z) 

I 

( a, 

b, 

c) 

J ( g)( 

z) 

, 

p 

, 

p 

, p 

 

zf 

( ) 

p z 

 

 

I 

 

, p 

 

( a, 

b, 

c) 

J 

, p 

 

I ( a, 

b, 

c) 

g 

, 

p 

z I 

, 

I 

, 

p 

, 

p 

, p 

, p 

zf ( 

) 

p z 

 

 

z f ( ) 

( 

) 

p z 

I 

, 

( a, 

b, 

c) 

J 

, 

zf 

p 

p 

 

p z 

 

 

 

( z) 

I 

, 

p 

( a, 

b, 

c) 

J 

, pg( 

z) 

 

p 

( a, 

b, 

c) 

J 

, p 

( g)( 

z) 

 

 

, p 

( a, 

b, 

c) 

J 

, pg( 

z) 

(3.7) 

 

 

 

Since g( z) 

S 

 

, p( 

a, 

b, 

c; 

) 

( S) 

, by virtue 

 

 

of Theorem 4, we have J g)( 

z) 

S ( a, 

b, 

c; 

) . 

Let us now put 

, p( , 

p 

 

.


 

z I 

pI 

, 

p 

, 

p 

( a, 

b, 

c) 

J 

( a, 

b, 

c) 

J 

, p 

, p 

( g)( 

z) 

 

 

( g)( 

z) 

H( 

z) 

, 

where H( z) 

( 

z) 

( z U) 

for 

S 

. Then, by 

using the same techniques as in the proof of 

Theorem 3, we conclude from (3.6) and (3.7) that 

z( 

I 

pI 

, 

p 

, 

p 

( a, 

b, 

c) 

f ( z)) 

zq( 

z) 

q( 

z) 

( z) 

( z U) . (3.8) 

( a, 

b, 

c) 

g( 

z) 

pH( 

z) 

 

Hence, upon setting 

1 

Q( z) 

 

( z U) 

pH( 

z) 

 

in (3.8), if we apply Lemma 2, we obtain 

q( z) 

( z) 

( zU) , 

which yields 

J 

, p( f )( z) 

C 

, p( 

a, 

b, 

c; 

, 

) ( , 

S) . 

The proof of Theorem 6 is thus completed. 

Remark 2. 

(i) Putting a 0 and b c in the above 

results we obtain the corresponding results, 

p 

for the operator ; 

I , 

(ii) Putting b p 1 , a c and replacing by 

1, 

p 1in the above results, we 

obtain the corresponding results for the 

( , 

p) 

operator . 

4. ACKNOWLEDGMENTS 

z 

The authors thank the referees for their valuable 

suggestions to improve the paper. 

5. REFERENCES 

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ISSN: 0377 - 2969 


Citations 

Citations of Newly Elected Fellows of PAS 

The following two eminent scientists were elected Fellows of the Pakistan Academy of Sciences during 

2011: 

Dr. Rumina Hasan 

Dr. Rumina Hasan (MBBS, PhD, FRC Path) is 

currently a Professor in the Department of 

Pathology & Microbiology at the Aga Khan 

University (AKU), Karachi and Honorary 

Professor at the London School of Hygiene and 

Tropical Medicine UK. She was instrumental in 

establishing the clinical mycobacterial laboratory 

at AKU; one of the few laboratories, performing 

M. tuberculosis culture and drug sensitivity in 

Pakistan. Dr. Hasan's research interests include 

antimicrobial resistance. She conducted the 

baseline studies for determining M. tuberculosis 

genogroups prevalent in Pakistan and their 

relationship to drug resistance. This work is now 

being taken forward to explore genetic markers 

associated with pathogenesis amongst strains 

prevalent in this region. Her work has also 

addressed the issue of M. tuberculosis drug 

resistance at a community level in particular 

evaluation of risk factors. 

Dr. Hasan's research group has explored 

molecular basis of antimicrobial resistance in 

bacterial organisms other than tuberculosis and has 

worked towards development of systems to reduce 

spread of resistant organisms in nosocomial 

settings. The antimicrobial resistance work is 

being taken forward to investigate resistance in 

fungal organisms. Her work on antimicrobial 

resistance has led to the establishment of a 

national task force to address the issue of 

antimicrobial resistance at a national level. 

In addition to research activity, Dr. Hasan has 

been involved in training. She initiated clinical 

microbiology residency outside the armed forces; 

to date 9 residents have passed their FCPS 

professional exam. She has also supervised 3 PhD 

students as primary supervisor, 2 of whom have 

completed. 

Dr. Muhammad Iqbal 

Dr. Muhammad Iqbal is the Director General, 

Centre for Applied Molecular Biology, Lahore. 

Prior to this, he was the Chief Scientific Officer 

and Head, Food and Biotechnology Research 

Center at PCSlR, Labs Complex, Lahore. He is an 

internationally known scientist for his pioneering 

work towards the development of novel 

immobilization technique developed by using 

indigenous low-cost agro-waste materials. This 

innovative immobilization technique has now 

become an essential tool for biotechnological 

research in the field of fermentation, 

bioremediation and biosorption and is being used 

worldwide for the entrapment of microalgae, 

fungi, yeasts, bacteria and plant cells. 

Dr. Iqbal has published 76 research papers in 

top ranking international and national journals 

with cumulative Impact Factor of 101.45 (JCR- 

2009, USA). The importance of his work can also 

be judged by more than 900 world-wide citations 

by Scopus/web of Science. Three of his research 

papers have been ranked ISI 1 st and 5th among the 

top 10 most cited papers during last five years 

(Scopus/Science Direct). He has also been 

honored with Extraordinary Career Accomplishment 

Award Letter by Web of Science, USA. 

Dr. Iqbal obtained his BSc (Hons) in 1979 and 

MSc (2nd position) in 1980, both in Botany, from 

the University of Karachi and MPhil (1 st position) 

in Plant Physiology from Quaid-i-Azam 

University, Islamabad in 1985. His PhD is in 

Microbial Biotechnology from the University of 

Sheffield, UK in 1990. He is the recipient of three 

prestigious international Research awards; 

Overseas Research Students (ORS) Award, UK 

(1986-89) Alexander von Humboldt Foundation 

Fellow, Germany (1996-98); and Fulbright Senior 

Research Fellow, USA (2006-07). In national

64 Citations of Elected Fellows 

competition, he has won the S&T International 

Talent Scholarship for PhD and the Post-doctoral 

International Fellowship Award, UK (2002), both 

awarded by the Ministry of Science and 

Technology, Pakistan. 

Dr. Jqbal was awarded Tamgha-i-Imtiaz in 

2010 by the President of Pakistan. He is recipient 

of PAS Gold Medal (2008) in Botany and the 

1996 TWAS Young Scientist of the Year Award 

in Biology of the Third World Academy of 

Sciences, Italy. His research achievements have 

also been recognized by the Government of 

Pakistan through Research Productivity Awards 

for eight (8) consecutive years, since 2001. 

Recently, he has been placed in " A " category of 

Scientists of Pakistan (2010-2011), based on 

Impact Factor and Citations of his publications, by 

the Ministry of Science and Technology of 

Pakistan. 

Citations of Newly Elected Foreign Fellows of PAS 

The following two eminent scientists were elected Foreign Fellows of the Pakistan Academy of 

Sciences during 2011: 

Prof. Dr. Chunli Bai 

Chunli Bai is Professor of Chemistry and 

President of the Chinese Academy of Sciences 

(CAS) and President of the Graduate University of 

CAS with more than 34,000 graduate students. 

Prof. Chunli Bai graduated from Department 

of Chemistry, Peking University in 1978 and 

received his MS and PhD degrees from CAS 

Institute of Chemistry in 1981 and 1985, 

respectively. During 1985-1987, he was Research 

Associate at Caltech, USA for advanced studies, 

conducting research in the field of physical 

chemistry. Thereafter he continued his research at 

CAS Institute of Chemistry. From 1991 to 1992, 

he was a visiting professor at Tohoku University 

in Japan. 

Research areas of Prof. Bai involve the 

structure and properties of polymer catalysts, X- 

ray crystallography of organic compounds, 

molecular mechanics and EXAFS research on 

electro-conducting polymers. In mid-1980s, he 

shifted his research orientation to the field of 

scanning tunneling microscopy, molecular 

nanostructures, self-assembly, novel nanomaterials, 

molecular nano devices, and single 

molecule detection. 

Prof. Bai is one of the pioneers in the field of 

scanning probe microscopy and nanotechnology in 

China. In mid-1980s while the scanning probe 

microscope was not yet commercially available, 

he successfully designed and developed China's 

first atomic force microscope (AFM), scanning 

tunneling microscope (STM), low-temperature 

STM, UHV-STM, and ballistic electron emission 

microscopy (BEEM). Due to his creative 

contributions to the solutions of a series of 

technical problems, he has earned a number of 

patents of original innovations and applications. 

These achievements were the landmarks of SPM 

research in China, leading to the earliest 

technological tools in the country for manipulating 

single atoms and molecules and characterizing 

surface and interface in the nano-scale world. 

Prof. Bai has been instrumental in furthering 

China's nanoscience and nanotechnology research 

both as a scientist and a policy-maker. As Chief 

Scientist of National Steering Committee for 

Nanoscience and Related Technology, he initiated 

and coordinated a number of national key projects 

about Nano S&T. He is the founding Director and 

Council Chairman of the National Center for 

Nanoscience and Technology, China. 

Prof. Bai has more than 350 scientific 

publications in refereed journals and has authored 

12 monographs and several book chapters in the 

field. He has won more than 20 prestigious awards 

and prizes for his academic achievements. He was 

elected a member of CAS and a Fellow of the 

Academy of Sciences for the Developing World 

(TWAS) in 1997. He is also Foreign Associate of 

the US National Academy of Sciences, Foreign 

Member of Russian Academy of Sciences, 

Member of the German National Academy of 

Science and Engineering (acatech.de) and 

Honorary Fellow of the Indian Academy of

Citations of Elected Fellows 65 

Sciences, the Royal Society of Chemistry, UK and 

the Chemical Research Society of India. Also, he 

has been honorary doctor or named fellow in 

several universities of USA, UK, Sweden, 

Demark, Russia, Australia, etc. He is recipient of 

the UNESCO first medal "for contributions to the 

development of nanoscience and nanotechnology" 

(shared with Nobel Laureate Professor Zhores 

Ivanovich Alferov); International Medal of the 

Society of Chemical Industry, London; and TWAS 

2002 Lecture Medal in Chemical Sciences. 

Because of his meritorious services, he is Vice 

President of TWAS, President of Federation of 

Asian Chemical Societies, President of Chinese 

Chemical Society, Honorary President of Chinese 

Society of Micro-Nano Technology and 

(CSMNT), Vice President of the China 

Association for Science and Technology, and Vice 

President of the Asia-Pacific Academy of 

Materials. 

Dr. G. Sarwar Gilani 

Dr. G. Sarwar Gilani did his MSc in Agricultural 

Chemistry from University of Peshawar; MSc and 

PhD in Nutritional Sciences from University of 

Saskatchewan, Canada. After working as Postdoctoral 

Fellow at University of Alberta and as 

Research Advisor for Rapeseed Association of 

Canada, joined Bureau of Nutritional Sciences of 

Health Canada in 1977. Currently, he is a Senior 

Research Scientist in Nutrition Research Division 

of Health Canada, Ottawa in the area of safety, 

nutritional quality and health aspects of dietary 

proteins and associated minor bioactive 

components. Previously, he was Adjunct Professor 

at McGill University and Universite Laval and has 

supervised graduate students’ research at the 

University of Ottawa and the University of 

Toronto. 

Dr. Gilani has authored 100 research papers, 

18 reviews and 25 book chapters. Impact factor of 

his publications is 295 and number of citations of 

his publications is 1132. He is Senior Co-Editor of 

the American Oil Chemists’ Society’s book, 

Phytoestrogens and Health (2002). 

Dr. Gilani has made 147 scientific 

presentations at national and international 

meetings, has served on expert panels, and has 

organized and chaired several international 

symposia including those on functional 

foods/nutraceuticals, bioactive peptides, 

phytoestrogens, foods derived through 

biotechnology, and trans-isomer fatty acids (Trans 

fats). He presented papers at the Global Biobusiness 

Forum “Bio Asia”, held in India in 2006- 

2009, and since 2002 has served as Editor of Food 

Composition and Additive Section of Journal of 

Association of Official Analytical Communities 

International (AOACI). Dr. Gilani participated in 

2002 joint FAO/WHO Expert Consultation on 

Protein and Amino Acid Requirements for 

Humans. Also, he was Scientific Advisor to 

FAO/WHO expert committees on Protein Quality 

Consultation on the assessment of nutritional 

requirements of infant formula. He has contributed 

to the Codex Alimentarius Commission’s Food 

Standards Programme and has advised Federation 

of Asian Biotech Association, Hyderabad, India 

about consumption of safe biofortified crops in 

reducing malnutrition and risk of chronic diseases 

in developing countries. 

Dr. Gilani guided for updating of graduate 

courses and laboratory facilities in Food 

Science/Nutrition at Faculty of Agriculture at 

Pakistan’s Gomal University and assisted in 

designing surveys to assess nutritional status of 

local populations and prepared a feasibility report 

for increased production and utilization of healthy 

oilseeds in D.I. Khan. Also, he provided guidance 

for improving food safety laws and nutritional 

quality and implementing food policies regarding 

nutritional needs of infants, school children and 

pregnant women; and reviewed Food and 

Nutrition Section of Pakistan’s 9 th Five-Year Plan. 

Dr. Gilani advised National Agriculture 

Research Center, Islamabad on research projects 

concerning removal of antinutritional factors and 

nutritional quality improvements in food crops, 

development of a nutritionally balanced formula 

based on local ingredients and folate-fortification 

of foods in Pakistan. He collaborated with 

Government of Pakistan in proposing the 

establishment of an Institute of Food and Nutrition 

and assisted in capacity building of National 

Institute of Food Science and Technology, 

University of Agriculture, Faisalabad. Dr. Gilani 

advised Pakistan’s Ministry of Health, regarding 

establishment of a National Institute of Nutrition. 

Also, he collaborated with Pakistan’s National 

Commission on Biotechnology and COMSTECH 

on projects and issues related to food 

biotechnology.

66 Citations of Elected Fellows 

Dr. Gilani was on the Organizing Committee 

and International Advisory Committee of the 

FAO-sponsored international symposium on 

Dietary Protein for Human Health, held in March 

2011 in Auckland, New Zealand. In recognition of 

his professional contributions to Canadian and 

international community in 2003, Dr. Gilani was 

awarded the Commemorative Gold Medal.

Proceedings of the Pakistan Academy of Sciences 49 (1): 67 (2012) 


ISSN: 0377 - 2969 


Book Review 

Biofertiliser Handbook: Research, Production, Application 

By Dr. P. Bhattacharyya and Dr. HLS Tandon, pp. 190 + x. ISBN: 81-85116- 64-4 (2012). 

Fertiliser Development and Consultation Organisation, New Delhi 110 048, India, www.tandontech.net 

This handbook provides a comprehensive and indepth 

coverage on Biofertilisers, starting from 

theoretical concepts to practical applications. The 

book contains 13 chapters on various aspects of 

Biofertiliser Technology. The first three chapters 

deal with biofertiliser classification while 4 th 

chapter covers crop responses to biofertliser 

inputs. In 5 th and 6 th chapters, production 

technology and storage issues have been critically 

discussed. Chapters 7 and 8 deal with marketing 

and commercialization constraints of biofertilisers, 

while chapter 9 provides an insight on practical 

recommendations of different kinds of 

biofertilisers. In chapter 10, the authors have 

critically discussed the most important issue of 

quality control of biofertilisers. Research and 

development efforts on biofertilizers undertaken 

by Indian as well as by some other institutions 

have been elaborated in chapter 11. Overall, it is 

an excellent effort to provide comprehensive 

knowledge on Biofertiliser Technology for the 

interested readers, including students, researchers 

and commercial stakeholders. 

The biofertiliser concept is not a new one; but 

it never got its due position amongst agricultural 

inputs because of a number of biofertiser-specific 

factors, such as poor quality control, shelf life 

limitation, mechanism of action, and inconsistency 

in evoking plant responses to biofertilisers. 

Farmers also used to show reluctance in adapting 

this input in the presence of chemical fertilizers. 

However, the realization regarding soil 

degradation due to intensive agriculture with high 

input of chemical fertilizers and unprecedented 

price hike of chemical fertilizers, coupled with 

their availability issue, have forced the scientists/ 

researchers and farmers to use biofertilisers as 

supplements to chemical fertilizers. Moreover, 

during the last decade, substantial advancements 

have been made in the field of biofertiliser 

technology with respect to quality control, shelf 

life and mechanisms of action. During the last 

couple of years, numerous products have been 

marketed under the umbrella of biofertilisers with 

different claims. So this was the right time for 

writing up of such kind of a book to enhance 

awareness amongst researchers/ scientists and 

other stakeholders about biofertilisers. The most 

unique aspect of this book is that it covers almost 

every aspect of Biofertiliser Technology. 

I strongly believe that this book could be a 

useful asset for the academics as well as for 

commercial production purposes. I recommend the 

book for undergraduate and graduate students of 

agriculture, particularly for the students of soil 

science. 

Dr. Muhammad Arshad, T.I. 

Professor (Tenured) 

Director, Institute of Soil & Environmental Sciences 

University of Agriculture, Faisalabad, Pakistan 

P.S.: In Pakistan, this book is available with Pak Book Corporation, Lahore, Islamabad, Karachi 

www.pakbook.com

Proceedings of the Pakistan Academy of Sciences 

Instructions for Authors 

Purpose and Scope: Proceedings of the Pakistan Academy of Sciences is official journal of the Pakistan Academy 

of Sciences, published quarterly, in English. It publishes original research papers in Engineering Sciences & 

Technology, Life Sciences, Medical Sciences, and Physical Sciences. State-of-the-art reviews (~ 20 pages, supported 

by recent references) summarizing R&D in a particular area of science, especially in the context of Pakistan, and 

suggesting further R&D are also considered for publication. Manuscripts undergo double-blind review. Authors are 

not required to be Fellows or Members of the Pakistan Academy of Sciences or citizens of Pakistan. 

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The manuscripts may contain Abstract, Keywords, INTRODUCTION, MATERIALS AND METHODS, 

RESULTS, DISCUSSION (or RESULTS AND DISCUSSION), CONCLUSIONS, ACKNOWLEDGEMENTS 

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objectives of the investigation. 

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references. 

ACKNOWLEDGEMENTS: In a brief statement, acknowledge financial support and other assistance.

70 

REFERENCES: Cite references in the text (in font size 10) by number only in square brackets, e.g. “Brown et al 

[2] reported ...” or “... as previously described [3, 6-8]”, and list them in REFERENCES section, in the order of 

citation in the text, Tables and Figures (not alphabetically). Only published (and accepted for publication) journal 

articles, books, and book chapters qualify for REFERENCES. List of REFERENCES must be prepared as under: 

a. Journal Articles (Name of journals must be stated in full) 

1. Golding, I. Real time kinetics of gene activity in individual bacteria. Cell 123: 1025–1036 (2005). 

2. Bialek, W. & S. Setayeshgar. Cooperative sensitivity and noise in biochemical signaling. Physical Review 

Letters 100: 258–263 (2008). 

3. Kay, R.R. & C.R.L. Thompson. Forming patterns in development without morphogen gradients: scattered 

differentiation and sorting out. Cold Spring Harbor Perspectives in Biology 1: doi: 

10.1101/cshperspect.a001503 (2009). 

b. Books 

4. Luellen, W.R. Fine-Tuning Your Writing. Wise Owl Publishing Company, Madison, WI, USA (2001). 

5. Alon, U. & D.N. Wegner (Eds.). An Introduction to Systems Biology: Design Principles of Biological 

Circuits. Chapman & Hall/CRC, Boca Raton, FL, USA (2006). 

c. Book Chapters 

6. Sarnthein, M.S. & J.D. Stanford. Basal sauropodomorpha: historical and recent phylogenetic developments. 

In: The Northern North Atlantic: A Changing Environment. Schafer, P.R. & W. Schluter (Eds.), Springer, 

Berlin, Germany, p. 365–410 (2000). 

7. Smolen, J.E. & L.A. Boxer. Functions of Europhiles. In: Hematology, 4 th ed. Williams, W.J., E. Butler, & 

M.A. Litchman (Eds.), McGraw Hill, New York, USA, p. 103–101 (1991). 

Tables, with concise but comprehensive headings, on separate sheets, must be numbered according to the order of 

citation (like Table 1., Table 2.). Round off data to the nearest three significant digits. Provide essential explanatory 

footnotes, with superscript letters or symbols keyed to the data. Do not use vertical or horizontal lines, except for 

separating column heads from the data and at end of the Table. 

Figures may be printed in two sizes: (i) column width of 8.0 cm; or (ii) entire page width of 16.5 cm; number them 

as Fig. 1., Fig. 2., … in the order of citation in the text. Captions to Figures must be concise nut self-explanatory. 

Computer generated, laser printed line drawings are acceptable. Do not use lettering smaller than 9 points or 

unnecessarily large. Photographs, digital or glossy prints, must be of high quality. A scale bar should be provided 

on all photomicrographs. 

Declaration: Provide a declaration that the results are original; the same material is neither ‘in press’ nor under 

consideration elsewhere; approval of all authors has been obtained to submit the manuscript; and, in case the article 

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permission to reproduce copyrighted material from other sources. 

Reviewers: Authors may suggest four prospective reviewers, two local and two from a scientifically advanced 

country. 

For detailed information, visit website of the Pakistan Academy of Sciences, i.e., www.paspk.org 

Manuscript Submission: Soft copy of the manuscript in Microsoft (MS) Word (pdf files not acceptable), as Email 

attachment or on a CD, may be submitted to: 

Editor-in-Chief 


3-Constitution Avenue, G-5/2, Islamabad, Pakistan 

E-mail: pas.editor@gmail.com 

Tel: +92-51-9207140 

Website: www.paspk.org

OF THE PAKISTAN ACADEMY OF SCIENCES 

C O N T E N T S 

Volume 49, No. 1, March 2012 

Page 

Research Articles 

Physical Sciences 

On Regions of Variability of Some Differential Operators Implying Starlikeness 1 

– Sukhwinder Singh Billing 

Some New s-Hermite-Hadamard Type Inequalities for Differentiable Functions and Their Applications 9 

– Muhammad Muddassar, Muhammad I. Bhatti and Muhammad Iqbal 


– O.R. Sayed 

A Study on Subordination Results for Certain Subclasses of Analytic Functions defined by Convolution 25 

– M.K. Aouf, A.A. Shamandy, A.O. Mostafa and A.K. Wagdy 

Existence and Uniqueness for Solution of Differential Equation with Mixture of Integer and 

Fractional Derivative 33 

– Shayma Adil Murad, Rabha W. Ibrahim and Samir B. Hadid 

On Stability for a Class of Fractional Differential Equations 39 

– Rabha W. Ibrahim 

Some Inclusion Properties of p-Valent Meromorphic Functions defined by the Wright Generalized 

Hypergeometric Function 45 

– M.K. Aouf, A.O. Mostafa, A.M. Shahin and S.M. Madian 

Some Inclusion Properties of Certain Operators 53 

– M.K. Aouf, R.M. El-Ashwah and E.E. Ali 

Citations of Elected Fellows 63 

Book Review 67 

Instructions for Authors 69 

PAKISTAN ACADEMY OF SCIENCES, ISLAMABAD, PAKISTAN 

HEC Recognised, Category X 

Website: www.paspk.org

TITLE MARCH 2012 - Pakistan Academy of Sciences

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