NUMBER OF ZEROS OF A POLYNOMIAL IN A CLOSED DISC
Gulzar M.H., Zargar B.A and Manzoor A.W
Department of Mathematics, University of Kashmir, Hazratbal, Srinagar 190006
A R T I C L E I N F O A B S T R A C T
In this paper we find the number of zeros of a polynomial in a closed disc, when the real and imaginary parts of the coefficients of the polynomial are restricted to certain conditions.
INTRODUCTION
In connection with a generalization of the Enestom-Kakeya Theorem [3,4] which states that all the n zeros of an nth degree
polynomial
n
j j j
z
a
z
P
0
)
(
witha
n
a
n1
...
a
1
a
0
0
lie inz
1
, Gulzar [2] very recently proved the following result:Theorem A:Let
n
j j j
z
a
z
P
0
)
(
be a polynomial of degreen
withRe(
a
j)
j,
Im(
a
j)
j,
n
j
0
,
1
,
2
,...,
such that for some
,
;
0
n
1
,
0
n
1
and for somek
1,
k
2
1
;
1,
2
1
,
,
...
...
2 1
2
1 1
1
n n
n n
k
k
and
L
1
1
2
...
1
0
0,
M
1
1
2
...
1
0
0,
Then all the zeros of P(z) lie in
].
)
(
)
(
[
1
)
1
(
)
1
(
2 1
2 1
2 1
M
L
k
k
a
a
k
i
k
z
n nn n
n n
2. MainRESULTS
In this paper we prove the following result:
Theorem 1:Let
n
j j j
z
a
z
P
0
)
(
be a polynomial of degreen
withRe(
a
j)
j,
Im(
a
j)
j,
n
j
0
,
1
,
2
,...,
such that for some
,
;
0
n
1
,
0
n
1
and for somek
1,
k
2
1
;
1,
2
1
,
Article History:
Received 29th November, 2016
Received in revised form 30thDecember, 2016 Accepted 4th January, 2017
Published online 28th February, 2017
Key words:
Coefficients, Polynomial, Zeros.
ISSN: O: 2319-6475, ISSN: P: 2319 – 6505, Impact Factor: SJIF: 5.438
Available Online at www.journalijcar.org
Volume 6; Issue 2; February 2017; Page No. 2081-2086
Research Article
,
...
...
2 1
2
1 1
1
n n
n n
k
k
and
L
1
1
2
...
1
0
0,
M
1
1
2
...
1
0
0,
Then the number of zero of P(z) in 0
,
c
1
c
R
z
X
a
is less than or equal to
0
log
log
1
a
A
c
forR
1
and the number ofzeros of P(z) in 0
,
c
1
c
R
z
Y
a
is less than or equal to
0
log
log
1
a
B
c
forR
1
, where
1
[
1((
)
2(
)
1(
)
n n n
n n
n n n
n
R
R
k
k
a
X
2(
)
L
M
0
0}
,
Y
a
nR
n1
R
[
n
n
k
1((
n
n)
k
2(
n
n)
1(
)
2(
)
L
M
]
(
1
R
)(
0
0)
,
1
[
1((
)
2(
)
1(
)
n n n
n n
n n n
n
R
R
k
k
a
A
2(
)
L
M
]
,
[
1((
)
2(
)
1(
)
1
n n n n n nn
n
R
R
k
k
a
B
2(
)
L
M
]
(
1
R
)(
0
0)
.For particular values of the parameters, we get many interesting results from Theorem 1. For example, for R=1,
1
0
,
c
, we get the following result:Corollary 1:Let
n
j j j
z
a
z
P
0
)
(
be a polynomial of degreen
withRe(
a
j)
j,
Im(
a
j)
j,
n
j
0
,
1
,
2
,...,
such that for some
,
;
0
n
1
,
0
n
1
and for somek
1,
k
2
1
;
1,
2
1
,
,
...
...
2 1
2
1 1
1
n n
n n
k
k
and
L
1
1
2
...
1
0
0,
M
1
1
2
...
1
0
0,
Then the number of zero of P(z) in 0
z
,
0
1
X
a
is less than or equal to
0
log
1
log
1
a
A
, where
X
a
n
n
n
k
1((
n
n)
k
2(
n
n)
1(
)
2(
)
L
M
0
0 ,
A
a
n
n
n
k
1((
n
n)
k
2(
n
n)
1(
)
2(
)
L
M
.For
1
2
1
, we get the following result from Theorem 1:Corollary 2:Let
n
j j j
z
a
z
P
0
)
(
be a polynomial of degreen
withRe(
a
j)
j,
Im(
a
j)
j,
n
,
...
...
1 2
1 1
n n
n n
k
k
and
L
1
1
2
...
1
0
0,
M
1
1
2
...
1
0
0,
Then the number of zero of P(z) in 0
,
c
1
c
R
z
X
a
is less than or equal to
0
log
log
1
a
A
c
forR
1
and the number ofzeros of P(z) in 0
,
c
1
c
R
z
Y
a
is less than or equal to
0
log
log
1
a
B
c
forR
1
, where
X
a
nR
n1
R
n[
n
n
k
1((
n
n)
k
2(
n
n)
L
M
0
0}
,
[
1((
)
2(
)
1
n n n
n n
n n
n
R
R
k
k
a
Y
L
M
]
(
1
R
)(
0
0)
,
[
1((
)
2(
)
1
n n n
n n
n n n
n
R
R
k
k
a
A
L
M
]
,
[
1((
)
2(
)
1
n n n
n n
n n
n
R
R
k
k
a
B
L
M
]
(
1
R
)(
0
0)
.For 1 1 2
1
2
k
k
, we get the following result from Theorem 1:Corollary 3:Let
n
j j j
z
a
z
P
0
)
(
be a polynomial of degreen
withRe(
a
j)
j,
Im(
a
j)
j,
n
j
0
,
1
,
2
,...,
such that for some
,
;
0
n
1
,
0
n
1
,
,
...
...
1 1
n n
n n
and
L
1
1
2
...
1
0
0,
M
1
1
2
...
1
0
0,
Then the number of zero of P(z) in 0
,
c
1
c
R
z
X
a
is less than or equal to
0
log
log
1
a
A
c
forR
1
and the number ofzeros of P(z) in 0
,
c
1
c
R
z
Y
a
is less than or equal to
0
log
log
1
a
B
c
forR
1
, where
X
a
nR
n1
R
n[
n
n
L
M
0
0]
,
Y
a
nR
n1
R
[
n)
n
L
M
]
(
1
R
)(
0
0)
,A
a
nR
n1
R
n[
n
n
L
M
]
,
B
a
nR
n1
R
[
n
n
L
M
]
(
1
R
)(
0
0)
.Lemmas
For the proof of Theorem 2, we make use of the following lemmas:
Lemma 1:Let f(z) (not identically zero) be analytic for
z
R
,
f
(
0
)
0
andf
(
a
k)
0
,
k
1
,
2
,...,
n
. Then
n
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:Let f(z) be analytic ,
f
(
0
)
0
andf
(
z
)
M
forz
R
. Then the number of zeros of f(z) in
,
c
1
c
R
z
isless than or equal to
)
0
(
log
log
1
f
M
c
.Lemma 2 is a simple deduction from Lemma 1. 4. Proofs of Theorems
Proof of Theorem 1: Consider the polynomial
F
(
z
)
(
1
z
)
P
(
z
)
a
z
a
a
z
a
z
a
a
z
a
a
z
a
z
a
z
a
z
n n n n n n n n n)
(
)
(
...
)
(
)
...
)(
1
(
1 1 1 1 1 0 1 1 1
...
(
a
1
a
0)
z
a
0
a
nz
n1
(
k
1
1
)
nz
n
(
k
1
n
n1)
z
n
(
n1
n2)
z
n1...
(
1
1
)
z
10 0 1 1 1 2 1 2 1 2 1 2 0 1 1 1 1
}
)
(
...
)
(
)
1
(
)
(
...
)
1
(
)
{(
)
(
...
)
(
)
1
(
a
z
z
z
z
z
k
z
k
i
z
z
z
n n n n n
G
(
z
)
a
0, where
G
(
z
)
a
nz
n1
(
k
1
1
)
nz
n
(
k
1
n
n1)
z
n
(
n1
n2)
z
n1...
(
1
1
)
z
1
}
)
(
...
)
(
)
1
(
)
(
...
)
1
(
)
{(
)
(
...
)
(
)
1
(
0 1 1 1 2 1 2 1 2 1 2 0 1 1 1 1z
z
z
z
z
k
z
k
i
z
z
z
n n n n n
For
z
R
, we have, by using the hypothesis...
[
)
1
(
)
1
(
)
(
z
a
nz
n1
k
1 nz
n
k
2 nz
n
k
1 n
n1z
n
n1
n2z
n1
G
1
1
z
1
(
1
1
)
z
1
1z
...
1
0z
k
2
n
n1z
n
n1
n2z
n1...
1
2
z
1
(
2
1
)
z
1z
1
...
1
0z
]
a
nR
n1
(
1
k
1)
nR
n
(
1
k
2)
nR
n
[
k
1
n
n1R
n
n1
n2R
n1
...
1
1
1(
11
)
1
1 ...
1
0
R
R
R
}]
...
)
1
(
...
0 1 1 2 1 2 1 1 2 1 1 2R
R
R
R
R
k
n n n n n n
a
nR
n1
R
n[(
1
k
1)
n
(
1
k
2)
n
k
1
n
n1
n1
n2
...
1
1
(
1
1
)
1
...
1
0
]
...
)
1
(
...
0 1 2 2 1 2 1 1 2
k
n n n n
a
nR
n1
R
n[(
1
k
1)
n
(
1
k
2)
n
n1
k
1
n
n2
n1
...
]
...
)
1
(
...
0 1
2 1 2
1 2 2
1
nk
n n n
1
[
1((
)
2(
)
1(
)
n n n
n n
n n n
n
R
R
k
k
a
2(
)
L
M
0
0}
=X for
R
1
and forR
1
(
)
[
1((
)
2(
)
1(
)
1
n n n n n nn
n
R
R
k
k
a
z
G
2(
)
L
M
]
(
1
R
)(
0
0)
=YSince G (z) is analytic for
z
R
,
G
(
0
)
0
, it follows by Schwarz Lemma thatz
X
z
G
(
)
forR
1
andG
(
z
)
Y
z
forR
1
. Hence forR
1
,
F
(
z
)
a
0
G
(
z
)
0
.
)
(
0 0
z
X
a
z
G
a
if
X
a
z
0and for
R
1
F
(
z
)
0
if
Y
a
z
0 .This shows that F(z) and hence P(z) does not vanish in
X
a
z
0 forR
1
and inY
a
z
0 forR
1
. In other words allthe zeros of P(z) lie in
X
a
z
0 forR
1
and inY
a
z
0 forR
1
.Again , for
z
R
, it is easy to see as above that
F
(
z
)
a
nR
n1
R
n[
n
n
k
1((
n
n)
k
2(
n
n)
1(
)
2(
)
L
M
]
=A for
R
1
and
(
)
1
[
1((
)
2(
)
1(
)
n n n
n n
n n
n
R
R
k
k
a
z
F
2(
)
L
M
]
(
1
R
)(
0
0)
=B for
R
1
.Hence , by using Lemma 1, it follows that the number of zeros of F(z) and therefore P(z) in
1
,
0
c
c
R
z
X
a
is less than or equal to
0
log
log
1
a
A
c
forR
1
and the number of zeros of F(z) and therefore P(z) in1
,
0
c
c
R
z
Y
a
is less than or equal to
log
log
1
a
B
That completes the proof of Theorem 1.
References
1. L. V. Ahlfors, Complex Analysis, 3rd edition, Mc-Grawhill.
2. M. H. Gulzar,B.A. Zargar and A. W. Manzoor, Zeros of a Polynomial with Restricted Coefficients, International
Research Journal of Advanced Engineering and Science, Vol 3No.2 (2017).
3. M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer. Math.Soc.(1966).
4. Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, New York (2002).
