e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 8, Issue 8 (September 2013), PP. 53-61
www.ijerd.com 53 | Page
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
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such that
a
n
a
n1
...
a
1
a
0
0
.Then the number of zeros of P(z) in
2
1
z
does not exceed
0
log
2
log
1
1
a
a
n
.K. K. Dewan [2] generalized Theorem A to polynomials with complex coefficients and proved the following results:
Theorem B: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such thatRe(
a
j)
j,Im(
a
j)
jand
n
n1
...
1
0
0
.Then the number of zeros of P(z) in
2
1
z
does not exceed
0 0
log
2
log
1
1
a
n
j j
n
.
Theorem C: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n with complex coefficients such that for somereal
,
,
a
j,
j
0
,
1
,
2
,...,
n
2
arg
and
a
n
a
n1
...
a
1
a
0 .Then the number f zeros of P(z) in
2
1
www.ijerd.com 54 | Page
0
1
0
sin
2
)
1
sin
(cos
log
2
log
1
a
a
a
n
j j
n
.
The above results were further generalized by researchers in various ways. M. H. Gulzar[4,5,6] proved the following results:
Theorem D: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such thatRe(
a
j)
j,Im(
a
j)
jand
n
n1
...
,
k
1
...
1
0,for
k
1
,
0
1
,
0
n
.
Then the number of zeros of P(z) inz
,
0
1
does not exceed
0
0 0
0
0
(
)
2
2
)
)(
1
(
log
1
log
1
a
k
n
j j n
n
.
Theorem E: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such thatRe(
a
j)
j,Im(
a
j)
jand
n
n1
...
1
0,for some
0
,
0
1
,
,then the number f zeros of P(z) inz
,
0
1
, does not exceed
0
0 0
0
0
)
2
2
(
2
log
1
log
1
a
n
j j n
n
.
Theorem F: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n with complex coefficients such that for somereal
,
,
a
j,
j
0
,
1
,
2
,...,
n
2
arg
and
a
n
a
n1
...
a
1
a
0 ,for some
0
,
,then the number of zeros of P(z) inz
,
0
1
, does not exceed0 1
1
0
(cos
sin
1
)
sin
2
)
1
sin
)(cos
(
log
1
log
1
a
a
a
a
n
j j
n
.
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
0
)
(
j j j
z
a
z
P
be a polynomial of degree n such thatRe(
a
j)
j,Im(
a
j)
jandwww.ijerd.com 55 | Page
fork
1
,
0
1
,
0
n
.
Then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
does not exceed
0
0 0
0 0
1
]
2
)
(
2
)
)(
1
(
[
log
log
1
a
k
R
c
n
j j n
n
n
for
R
1
and
0
1 0
0 0 0
0
[
(
1
)(
)
(
)
2
]
log
log
1
a
k
R
a
c
n
j j n
n
for
R
1
.Remark 1: Taking R=1 and
1
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
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such that
a
n
a
n1
...
a
,
ka
a
1
...
a
1
a
0,for
k
1
,
0
1
,
0
n
.
Then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
does not exceed
0
0 0 0
1
)]
(
2
)
)(
1
(
[
log
log
1
a
a
a
a
a
a
k
a
a
R
c
n n
n
for
R
1
and
0
0 0 0
0
[
(
1
)(
)
(
)]
log
log
1
a
a
a
a
a
a
k
a
a
R
a
c
n
n
for
R
1
. Applying Theorem 1 to the polynomial –iP(z) , we get the following result:Theorem 2: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such thatRe(
a
j)
j,Im(
a
j)
jand
n
n1
...
,
k
1
...
1
0,for
k
1
,
0
1
,
0
n
.
Then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
does not exceed
0
0 0
0 0
1
]
2
)
(
2
)
)(
1
(
[
log
log
1
a
k
R
c
n
j j n
n
n
for
R
1
and
0
1 0
0 0
0
0
[
(
1
)(
)
(
)
2
]
log
log
1
a
k
R
a
c
n
j j n
n
www.ijerd.com 56 | Page
forR
1
.Taking
n
in Theorem 1, we get the following result :Corollary 2: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such thatRe(
a
j)
j,Im(
a
j)
jand
k
n
n1
...
1
0,for
k
1
,
0
1
,
0
n
.
Then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
does not exceed
0
0 0
0 0
1
]
2
)
(
2
)
(
[
log
log
1
a
k
R
c
n
j j n
n
n
for
R
1
and
0
0 0
0 0
0
[
(
)
(
)
2
]
log
log
1
a
k
R
a
c
n
j j n
n
for
R
1
.Theorem 3: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such thatRe(
a
j)
j,Im(
a
j)
jand
n
n1
...
1
0,for some
,
0
1
,
,then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
, does not exceed
0
0 0
0 0 1
]
2
2
)
(
[
log
log
1
a
R
c
n
j j n
n
n
for
R
1
and
0
0 0
0 0
0
[
(
)
2
]
log
log
1
a
R
a
c
n
j j n
n
for
R
1
.Remark 2: Taking R=1 and
1
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
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such that
a
n
a
n1
...
a
1
a
0,for some
,
0
1
.
Then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
, does not exceed
0
0 0
0 1
]
2
)
(
[
log
log
1
a
a
a
a
a
a
R
c
n n
n
www.ijerd.com 57 | Page
and
0
0 0 0
0
[
(
)
]
log
log
1
a
a
a
a
a
a
R
a
c
n
n
Applying Theorem 3 to the polynomial –iP(z) , we get the following result:
Theorem 4: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such thatRe(
a
j)
j,Im(
a
j)
jand
n
n1
...
1
0,for some
0
,
0
1
.
Then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
, does not exceed
0
0 0
0 0 1
]
2
2
)
(
[
log
log
1
a
R
c
n
j j n
n
n
for
R
1
and
0
0 0
0 0
0
[
(
)
2
]
log
log
1
a
R
a
c
n
j j n
n
for
R
1
. Taking
(
k
1
)
n,
k
1
in Corollary 3 , we get the following result:Corollary 4: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such thatRe(
a
j)
j,Im(
a
j)
jand
k
n
n1
...
1
0,for some
,
0
1
.
Then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
, does not exceed
0
0 0
0 0 1
]
2
2
)
(
)
(
[
log
log
1
a
k
R
c
n
j j n
n
n
for
R
1
and
0
0 0
0 0
0
[
(
)
(
)
2
]
log
log
1
a
k
R
a
c
n
j j n
n
for
R
1
.Theorem 5: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n such that
a
n
a
n1
...
a
1
a
0for some
,
0
1
.
Then the number of zeros of P(z) in
(
R
0
,
c
1
)
c
R
z
, does not exceed
log
1
[
{(
)(cos
sin
1
)
(cos
sin
1
)
2
}]
log
1
0 0
1 0
n n
a
R
a
www.ijerd.com 58 | Page
forR
1
and
log
1
[
{(
)(cos
sin
1
)
(cos
sin
1
)
}]
log
1
0 0
0 0
a
a
a
R
a
a
c
n
for
R
1
. For different values of the parametersk
,
,
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 andn
k
a
f
(
k)
0
,
1
,
2
,...,
, then
nj j
i
a
R
f
d
f
1 2
0
log
(Re
log
(
0
)
log
2
1
.Lemma 1 is the famous Jensen’s theorem (see page 208 of [1]).
Lemma 2: If f(z) is analytic and
f
(
z
)
M
(
r
)
inz
r
, then the number of zeros of f(z) in
,
c
1
c
r
z
does not exceed
)
0
(
)
(
log
log
1
f
r
M
c
.Lemma 2 is a simple deduction from Lemma 1.
Lemma 3: Let
0
)
(
j j j
z
a
z
P
be a polynomial o f degree n with complex coefficients such that for somereal
,
,,
0
,
2
arg
a
j
j
n
anda
j
a
j1,
0
j
n
,
then any t>0,
ta
j
a
j1
(
t
a
j
a
j1)
cos
(
t
a
j
a
j1)
sin
. 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
)(
...
1 0)
1
1
z
a
z
a
a
z
a
z
n nn
n
a
z
1(
a
na
n 1)
z
n...
(
a
1a
0)
z
a
0 nn
a
nz
n1
a
0
(
n
n1)
z
n
...
(
1
)
z
1
n
j
j j
j
z
i
z
z
z
k
k
0
1 0
0 0
1
1 2 1
1
)
(
)]
(
)
[(
...
)
(
]
)
1
(
)
[(
For
z
R
, we have by using the hypothesisF
(
z
)
a
nR
n 1
a
0 +
n
n1R
n
...
1
R
1
k
1R
(
k
1
)
R
1
2R
1
...
1
0R
(
1
)
0R
nj
j j 1
1
)
www.ijerd.com 59 | Page
0[
1...
1
1(
1
)
1
k
k
R
a
R
a
n nn n
n
...
(
1
)
2
]
11 0
0 0
1 2
1
nj j
n
a
nR
n
a
0
R
n
n
k
0
0
0
0
n 1)
(
)
)(
1
(
[
2
]
1
1
nj j
R
1[
n
n
(
k
1
)(
)
(
0
0)
2
0n
2
]
0
n jj
forR
1
and
nj j n
n
k
R
a
z
F
1 0
0 0 0
0
[
(
1
)(
)
(
)
2
]
)
(
for
R
1
. Therefore , by Lemma 3, it follows that the number of zeros of F(z) and henceP(z) in
(
R
0
,
c
0
)
c
R
z
does not exceed
0
0 0
0 0
1
2
)
(
2
)
)(
1
(
[
log
log
1
a
k
R
c
n
j j n
n
n
for
R
1
and
0
1 0 0 0 0
0
[
(
1
)(
)
(
)
2
log
log
1
a
k
R
a
c
n
j j n
n
for
R
1
. That proves Theorem 1.Proof of Theorem 3: Consider the polynomial F(z) =(1-z)P(z)
(
1
)(
...
1 0)
1
1
z
a
z
a
a
z
a
z
n nn
n
a
nz
n1
(
a
n
a
n1)
z
n
...
(
a
1
a
0)
z
a
0
a
z
1a
0z
(
n n 1)
z
n...
(
2 1)
z
2 nn
n
nj
j j
j
z
i
z
0
1 0
0 0
1
)
(
)]
(
)
[(
.For
z
R
, we have by using the hypothesis
F
(
z
)
a
nR
n1
a
0 +
R
n
n
n1R
n
...
2
1R
2
1
0R
+ j n
j
j
j
R
R
0
1
0
(
)
)
1
(
R
n1[
n
0
n
n1
...
2
1
1
0
nj j 0 0
2
)
1
www.ijerd.com 60 | Page
[
(
)
2
2
]
0 0
0 0
1
nj j n
n n
R
forR
1
and
nj j n
n
R
a
z
F
0 0 0 0
0
[
(
)
2
]
)
(
forR
1
.Therefore , by Lemma 2, it follows that the number of zeros of F(z) and hence
P(z) in
(
R
0
,
c
0
)
c
R
z
does not exceed
0
0 0
0 0 1
]
2
2
)
(
[
log
log
1
a
R
c
n
j j n
n
n
for
R
1
and
0
0 0
0 0
0
[
(
)
2
]
log
log
1
a
R
a
c
n
j j n
n
for
R
1
. That proves Theorem 3.Proof of Theorem 5: Consider the polynomial F(z) =(1-z)P(z)
(
1
)(
...
1 0)
1
1
z
a
z
a
a
z
a
z
n nn
n
a
nz
n1
(
a
n
a
n1)
z
n
...
(
a
1
a
0)
z
a
0
a
z
1a
0z
(
a
na
n 1)
z
n...
(
a
2a
1)
z
2 nn
n
[(
a
1
a
0)
(
a
0
a
0)]
z
.For
z
R
, we have by using the hypothesis and Lemma 30 1
)
(
z
a
R
a
F
n n
+
R
n
a
n
a
n1R
n
...
a
2
a
1R
2
a
1
a
0R
+
(
1
)
0R
a
nR
n1
a
0
R
n
[(
a
n
a
n1)
cos
(
a
n
a
n1)
sin
]
R
n
R
a
a
a
a
R
a
R
a
a
a
a
]
sin
)
(
cos
)
[(
)
1
(
]
sin
)
(
cos
)
[(
...
0 1 0
1
2 0 2
1 2 1
2
R
n1[(
a
n)(cos
sin
1
)
0(cos
sin
1
)
2
a
0]
for
R
1
and
a
0
R
[(
a
n)(cos
sin
1
)
a
0(cos
sin
1
)
a
0]
for
R
1
. Hence , by Lemma 2, it follows that the number of zeros of F(z) and therefore P(z)in
(
R
0
,
c
0
)
c
R
z
does not exceed}]
2
)
1
sin
(cos
)
1
sin
)(cos
{(
[
1
log
log
1
0 0
1 0
n n
a
R
a
www.ijerd.com 61 | Page
forR
1
and
log
1
[
{(
)(cos
sin
1
)
(cos
sin
1
)
}]
log
1
0 0
0 0
a
a
a
R
a
a
c
n
for
R
1
. That completes the proof of Theorem 5.REFERENCES
[1]. L.V.Ahlfors, Complex Analysis, 3rd 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.
