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International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 8, Issue 8 (September 2013), PP. 53-61
On The Zeros of a Polynomial in a Given Circle
M. H. Gulzar
Department of Mathematics University of Kashmir, Srinagar 190006
Abstract:- In this paper we discuss the problem of finding the number of zeros of a polynomial in a given circle
when the coefficients of the polynomial or their real or imaginary parts are restricted to certain conditions. Our
results in this direction generalize some well- known results in the theory of the distribution of zeros of
polynomials.
Mathematics Subject Classification: 30C10, 30C15
Keywords and phrases:- Coefficient, Polynomial, Zero.
I. INTRODUCTION AND STATEMENT OF RESULTS
In the literature many results have been proved on the number of zeros of a polynomial in a given circle.
In this direction Q. G. Mohammad [6] has proved the following result:
Theorem A: Let
( )
j
a z P be a polynomial o f degree n such that
j
j0
z
an an
1
Then the number of zeros of P(z) in
a
1
...... a1
0
1
2
1 an log
log 2 a
0
z does not exceed
.
0
K. K. Dewan [2] generalized Theorem A to polynomials with complex coefficients and proved the following
results:
Theorem B: Let
j
P( z)
a be a polynomial o f degree n such that a j ) j Im( a ) and
Then the number of zeros of P(z) in
j z
j0
n n1
...... 1
0
1
1
log 2
1
2
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.
0
z does not exceed
log
n
j0
a
n
0
j
.
.
Re( , j j
Theorem C: Let
j
P( z)
a be a polynomial o f degree n with complex coefficients such that for some
real , ,
and
j0
j z
arg a j , j 0,
1,
2,......,
n
2
an an
1 ......
a1
a0
Then the number f zeros of P(z) in
.
1
2
z does not exceed

1
log 2
log
a
n 1
n (cos sin 1)
2sin
j 0
a
0
On The Zeros of a Polynomial in a Given Circle
The above results were further generalized by researchers in various ways.
M. H. Gulzar[4,5,6] proved the following results:
Theorem D: Let
j
P( z)
a be a polynomial o f degree n such that a j ) j
j z
j0
n n1
, k
1
...... .......
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a
j
.
1
Im( a ) and
Re( , j j
for k 1, 0
1,
0 n.
Then the number of zeros of P(z) in 1 0 , z does not exceed
1
log
1
log
n n ( k 1)(
) 2
0
a0
( 0 0 ) 2
j
j0
.
Theorem E: Let
P( z)
a
j
be a polynomial o f degree n such that a j ) j Im( a ) and
j z
j0
n n1
...... 1
0 ,
0
,
Re( , j j
for some 0, 0
1,
,then the number f zeros of P(z) in z ,
0 1,
does not exceed
2
n
1
log
1
log
n
( 0 0 ) 2
0
a0
2
j
j0
.
Theorem F: Let
P( z)
a
j
be a polynomial o f degree n with complex coefficients such that for some
real , ,
and
j0
j z
arg a j , j 0,
1,
2,......,
n
2
,
an an
1 ......
a1
a0
for some 0,
,then the number of zeros of P(z) in z ,
0 1,
does not exceed
n 1
n )(cos sin 1)
2sin
j 1
( a
1
log
1
log
a0
a j a0
(cos
sin 1)
.
The aim of this paper is to find a bound for the number of zeros of P(z) in a circle of radius not
necessarily less than 1. In fact , we are going to prove the following results:
Theorem 1: Let
( )
j
a z P be a polynomial of degree n such that a j ) j Im( a ) and
j
j0
z
n n1
, k
1
...... .......
1
n
Re( , j j
0
,
n

On The Zeros of a Polynomial in a Given Circle
for k 1, 0
1,
0 n.
Then the number of zeros of P(z) in z ( R 0,
c 1)
does not exceed
and
R
1
log
log c
1
log
log c
a
0
n1
[
n n
0
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0
0
R
c
0
n
j0
( k 1)(
) 2
( ) 2 ]
a
0
for R 1
n
n n ( k 1)(
)
( 0 0 ) 0 0 2
j1
R[
]
1
a
for R 1.
Remark 1: Taking R=1 and c in Theorem 1, it reduces to Theorem D .
If the coefficients a j are real i.e. j 0,
j
, then we get the following result from Theorem 1:
Corollary 1: Let
j
P( z)
a be a polynomial o f degree n such that
j z
j0
an an
1 a
, ka
a
1
...... ....... a a
for k 1, 0
1,
0 n.
Then the number of zeros of P(z) in z ( R 0,
c 1)
does not exceed
and
1 R
log
log c
1
log
log c
a
0
n1
[ a
n
a
n
( k 1)(
a
a
0
a
0
) 2 a
0
0
1
( a
0
for R 1
R[
a a ( k 1)(
a a ) a
( a
n
n
a
0
0
,
R
c
a )]
a
for R 1.
Applying Theorem 1 to the polynomial –iP(z) , we get the following result:
Theorem 2: Let
j
P( z)
a be a polynomial o f degree n such that a j ) j
j z
j0
n n1
, k
1
...... .......
1
0
0
)]
Re( , j j
0
,
j
j
Im( a ) and
for k 1, 0
1,
0 n.
Then the number of zeros of P(z) in z ( R 0,
c 1)
does not exceed
and
R
1
log
log c
1
log
log c
a
0
n1
[
n n
0
0
0
R
c
0
n
j0
( k 1)(
) 2
( ) 2 ]
a
0
for R 1
n
n n ( k 1)(
) 0 0
( 0 0 ) 2
j1
R[
]
a
j
j

Taking n in Theorem 1, we get the following result :
On The Zeros of a Polynomial in a Given Circle
for R 1.
Corollary 2: Let
j
P( z)
a be a polynomial o f degree n such that a j ) j
j z
j0
n n1
....... 1
0
k ,
Im( a ) and
Re( , j j
for k 1, 0
1,
0 n.
Then the number of zeros of P(z) in z ( R 0,
c 1)
does not exceed
and
R
1
log
log c
1
log
log c
n1
a
0
0
0
0
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0
n
j0
[ k(
) 2
( ) 2 ]
n
n
a
0
for R 1
R[
k(
)
( ) 2 ]
n n 0 0 0
j0
a
n
j
j
R
c
for R 1.
Theorem 3: Let
j
P( z)
a be a polynomial o f degree n such that a j ) j
j z
j0
n n1
...... 1
0 ,
Im( a ) and
Re( , j j
for some , 0 1,
,then the number of zeros of P(z) in z ( R 0,
c 1)
, does not exceed
and
R
1
log
log c
1
log
log c
n1
a
0
0
0
0
R
c
0
n
j0
[
( ) 2
2 ]
n
n
a
0
for R 1
R[
( ) 2 ]
1
n n 0 0 0
j0
a
for R 1.
Remark 2: Taking R=1 and c in Theorem 3, it reduces to Theorem E .
If the coefficients a j are real i.e. j 0,
j
, then we get the following result from Theorem 3:
Corollary 3: Let
j
( ) a z P be a polynomial o f degree n such that
j0
j z
,
an an
1 ...... a1
a0
for some , 0 1.
Then the number of zeros of P(z) in z ( R 0,
c 1)
, does not exceed
1 R
log
log c
n1
[ a
n
a
n
a
0
( a
0
R
c
a ) 2 a
0
0
]
n
j
j
for R
1

and
1
log
log c
a
0
R[
a a
( a
n
a
On The Zeros of a Polynomial in a Given Circle
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0
n
0
a ) a
Applying Theorem 3 to the polynomial –iP(z) , we get the following result:
Theorem 4: Let
P( z)
a
j
be a polynomial o f degree n such that a j ) j
j z
j0
n n1
...... 1
0 ,
0
0
]
Im( a ) and
Re( , j j
for some 0, 0 1.
Then the number of zeros of P(z) in z ( R 0,
c 1)
, does not exceed
and
R
1
log
log c
1
log
log c
n1
a
0
0
0
0
0
R
c
n
j0
[
( ) 2 2 ]
n
n
a
0
for R 1
R[
( ) 2 ]
n n 0 0 0
j0
for 1
Taking ( 1)
, k 1
in Corollary 3 , we get the following result:
k n
a
R .
Corollary 4: Let
j
P( z)
a be a polynomial o f degree n such that a j ) j
j z
j0
n n1
...... 1
0
k ,
n
j
j
Im( a ) and
Re( , j j
for some , 0 1.
Then the number of zeros of P(z) in z ( R 0,
c 1)
, does not exceed
and
R
1
log
log c
1
log
log c
n1
a
0
0
0
0
0
R
c
n
j0
[ k(
)
( ) 2
2 ]
n
n
a
0
for R 1
R[
k(
)
( ) 2 ]
n n
0 0 0
j0
a
n
for R 1.
Theorem 5: Let
j
( ) a z P be a polynomial o f degree n such that
j0
j z
an an
1 ...... a1
a0
for some , 0 1.
Then the number of zeros of P(z) in z ( R 0,
c 1)
, does not exceed
1 1 n
log [ R
log c
a
0
1
{( a )(cos
sin 1)
(cos
sin 1)
2
}]
n
R
c
0
j
j
0

and
On The Zeros of a Polynomial in a Given Circle
for R 1
1 1
log [ a0
R{(
an
)(cos
sin 1)
a0
(cos
sin 1)
a0
}]
log c a
0
for R 1.
For different values of the parameters k , , in the above results, we get many other interesting results.
II. LEMMAS
For the proofs of the above results we need the following results:
Lemma 1: If f(z) is analytic in z R ,but not identically zero, f(0) 0 and
f ( ak
) 0,
k 1,
2,......,
n , then
1
2
2
n
i
log f (Re d
log f ( 0)
log
0
j1 a j
Lemma 1 is the famous Jensen’s theorem (see page 208 of [1]).
Lemma 2: If f(z) is analytic and ( z)
M(
r)
does not exceed
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R
.
r
f in z r , then the number of zeros of f(z) in z , c 1
c
1 M ( r)
log .
log c f ( 0)
Lemma 2 is a simple deduction from Lemma 1.
Lemma 3: Let
( )
j
a z P be a polynomial o f degree n with complex coefficients such that for some
j0
j z
a j
2
a j j 1
ta a t a a ) cos
( t a a ) sin .
real , , arg , 0 j n,
and a , 0 j n,
then any t>0,
j j1
( j j1
j j1
Lemma 3 is due to Govil and Rahman [4].
III. PROOFS OF THEOREMS
Proof of Theorem 1: Consider the polynomial
F(z) =(1-z)P(z)
1 z)(
a
n
z a
n1
z ...... a z a )
( n n
1
1 0
n1
an
z an
an1
a
z
n
n1
n
( ) z ...... ( a a ) z a
a
0
[( k
n
( ) z ...... ( ) z
1
n
n1
) ( k 1)
] z
[(
) ( )] z i
1
For z R , we have by using the hypothesis
F
n1
( z)
an
R a0
0
0
+
0
1
(
n
j0
0
1
j
1
(
0
2
j1
) z
) z
1
1
j
......
R
R
R k
n
1
n n1
...... 1
1
1
( k 1)
R
1
2
R ...... 1
0
R ( 1
) 0
R
n
j1
(
j
j1
)

and
On The Zeros of a Polynomial in a Given Circle
n1
n
an R a0
R [ n
n1
......
1
k
1
( k 1)
1 2 ...... 1 0 ( 1 ) 0 0 2
n
n
]
n1
n
an R a0
R [ n ( k 1)(
)
( 0
0)
0
0
n
2
1
1
n
j
j
]
n1
R n
n ( k 1)(
[ )
(
) 2
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0
0
0
n
2 j
0
1
j1
j
] for R 1
F(
z)
a R[
( k 1)(
)
( ) 2 ]
0
n n
for 1
Therefore , by Lemma 3, it follows that the number of zeros of F(z) and hence
R
c
P(z) in z ( R 0,
c 0)
does not exceed
and
1
log
log c
R
1
log
log c
a
0
n1
[
n n
0
0
0
0
0
0
R .
0
0
n
j0
( k 1)(
) 2
(
) 2
0
a
for R 1
n
n
n ( k 1)(
)
( 0
0 ) 0 0 2
j1
R[
That proves Theorem 1.
Proof of Theorem 3: Consider the polynomial
F(z) =(1-z)P(z)
n
n1
1 z)(
a z a z ......
( n n
1
a1z
a0
n1
an
z an
an1
n
( ) z ...... ( a a ) z a
n1
n
an
z a0
z n n1
1
a
)
0
0
for R 1.
n
( ) z ...... ( ) z
n
j0
j
[(
) (
)] z i ( ) z .
1
For z R , we have by using the hypothesis
F
n1
( z)
an
R a0
0
0
0
j
j1
2
1
2
j
n
j1
n
n
2
+ R
R
R R
+ ( 1
) 0 R ( j j
j0
n
n n
1 ...... 2 1
1 0
1
) R
n1
R n 0
n
n1
[ ......
n
j0
( 1
) 2 ]
0
j
j
2
1
1
0
j
j
j

and
On The Zeros of a Polynomial in a Given Circle
n1
R [
( ) 2
2 ] for R 1
n
n
0
F(
z)
a R[
( ) 2 ] for R 1.
0
n
n
0
0
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0
0
0
n
j0
n
j 0
Therefore , by Lemma 2, it follows that the number of zeros of F(z) and hence
R
c
P(z) in z ( R 0,
c 0)
does not exceed
and
R
1
log
log c
1
log
log c
n1
a
0
0
0
0
0
j
j
n
j0
[
( ) 2
2 ]
n
n
a
0
for R 1
R[
( ) 2 ]
That proves Theorem 3.
Proof of Theorem 5: Consider the polynomial
F(z) =(1-z)P(z)
n
n1
1 z)(
a z a z ......
( n n
1
a1z
a0
n1
an
z an
an1
n n 0 0 0
j0
n
( ) z ...... ( a a ) z a
n1
n
an
z a0
z an
an1
[( a 1
a0
) ( a0
a0
)] z .
a
1
)
0
0
for R 1.
n
( ) z ...... ( a a ) z
For z R , we have by using the hypothesis and Lemma 3
F
n1
( z)
an
R a0
+ ( 1
)
R
0
n
n
2
+ R a a R a a R a a R
n n
1 ...... 2 1 1 0
n1
n
a n R a0
R [(
an
an1
) cos
( an
an1
) sin
]
...... [( a
[( a
1
2
a
0
a ) cos
( a
1
) cos
( a
1
2
a
a ) sin ]
R
0
1
) sin ]
R
2
2
1
( 1
) a
n1
R [( an
)(cos
sin
1)
0
(cos
sin
1)
2a0
]
for R 1
and
a0 R[(
an
)(cos
sin
1)
a0
(cos
sin
1)
a0
]
for R 1.
Hence , by Lemma 2, it follows that the number of zeros of F(z) and therefore P(z)
R
in z ( R 0,
c 0)
does not exceed
c
1 1 n1
log [ R {( an
)(cos
sin 1)
0 (cos
sin 1)
2
0 }]
log c
a
0
0
n
2
j
R
2
j
R
n

and
On The Zeros of a Polynomial in a Given Circle
for R 1
1 1
log [ a0
R{(
an
)(cos
sin 1)
a0
(cos
sin 1)
a0
}]
log c a
0
That completes the proof of Theorem 5.
for R 1.
REFERENCES
[1]. L.V.Ahlfors, Complex Analysis, 3 rd edition, Mc-Grawhill
[2]. K. K. Dewan, Extremal Properties and Coefficient estimates for Polynomials with restricted Zeros and
on location of Zeros of Polynomials, Ph.D. Thesis, IIT Delhi, 1980.
[3]. N. K. Govil and Q. I. Rahman, On the Enestrom- Kakeya Theorem, Tohoku Math. J. 20(1968),126-
136.
[4]. M. H. Gulzar, On the Zeros of a Polynomial inside the Unit Disk, IOSR Journal of Engineering, Vol.3,
Issue 1(Jan.2013) IIV3II, 50-59.
[5]. M. H. Gulzar, On the Number of Zeros of a Polynomial in a Prescribed Region, Research Journal of
Pure Algebra-2(2), 2012, 35-49.
[6]. M. H. Gulzar, On the Zeros of a Polynomial, International Journal of Scientific and Research
Publications, Vol. 2, Issue 7,July 2012,1-7.
[7]. Q. G. Mohammad, On the Zeros of Polynomials, Amer. Math. Monthly 72(1965), 631-633.
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