Applied Mathematics, 2011, 2, 1356-1358
doi:10.4236/am.2011.211189 Published Online November 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
On the Zeros of a Polynomial
Mohammad Syed Pukhta
Division of Agricultural Engineering, Sher-e-Kashmir University of Agricultural Sc iences
& Technology of Kashmir, Srinagar, India
E-mail: mspukhta_67@yahoo.co.in
Received February 27, 2011; revised May 25, 2011; accepted J une 3, 2011
Abstract
In this paper we consider the problem of finding the estimate of maximum number of zeros in a prescribed
region and the results which we obtain generalizes and improves upon some well known results.
Keywords: Polynomial, Zeros, Complex Number, Prescribed Region
1. Introduction
Let be a polynomial of degree n such
that
0
ni
i
i
pz az
110
0
nn
aa aa
then according to a well known result of Enstrom and
Kakeya, the polynomial
,pz does not vanish in
1.z concerning the number of zeros of the polyno-
mial in the region 1,
2
z the following result is due to
Mohammad [1].
Theorem A. Let be a polynomial of
degree n such that
0
ni
i
i
pz az
110
0,
nn
aa aa
then the number of zeros of
pz in 1,
2
z does not
exceed
1
1log
log 2
n
o
a
a
.
Dewan [2] generalized Theorem A to the polynomials
with complex coefficients and obtained the following
result.
Theorem B. If is a polynomial of de-
0
ni
i
i
pz az
gree n with complex coefficients such that
π
arg,0,1, 2,,
2
i
ai
11
,
nn
aa aa
0
then the number of zeros of in
pz 1
2
z does not
exceed
1
0
0
cossin12sin
1log
log 2
n
ni
i
aa
a
.
Theorem C. Let be a polynomial of
0
ni
i
i
pz az
degree n with complex coefficients. If
Re, Im,
ii ii
aa
for and 0,1, ,in
110
0,
nn
then the number of zeros
of
pz in 1
2
z
does not exceed
0
0
1
1log
log 2
n
ni
i
a
.
In this paper we generalize Theorem B and Theorem C
under less restrictive conditions on the coefficients,
which also improve upon them. More precisely, we
prove the following.
Theorem 1. Let be a polynomial of
0
ni
i
i
pz az
degree n with complex coefficients, such that
π
arg,0,1,2,,.
2
i
ai
n for some real
and
110
0
nn
aa aa
n for some real
and
then the number of zeros of in
pz ,z
does not
exceed