TITLE MARCH 2012 - Pakistan Academy of Sciences

Proceedings of the Pakistan Academy of Sciences 49 (1): 53-61 (2012)
Copyright © Pakistan Academy of Sciences
ISSN: 0377 - 2969
Pakistan Academy of Sciences
Original Article
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
1. INTRODUCTION
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