International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 13, Issue 3 (March 2017), PP.01-08
A Ring-Shaped Region Containing All or A Specific Number of The Zeros of A Polynomial M. H. Gulzar 1 , A. W. Manzoor 2 Department Of Mathematics, University Of Kashmir, Srinagar
ABSTRACT: According to a Cauchy’s classical result all the zeros of a polynomial P( z )
n
a j 0
degree n lie in
z 1 A , where A max 0 j n 1
aj
j
z j of
. In this paper we find a ring-shaped region containing
an
all or a specific number of zeros of P(z). Mathematics Subject Classification: 30C10, 30C15. Keywords and Phrases: Polynomial, Region, Zeros.
I.
INTRODUCTION
A classical result giving a region containing all the zeros of a polynomial is the following known as Cauchy’s Theorem [4]: n
Theorem A. All the zeros of the polynomial
P( z ) a j z j of degree n lie in the circle j 0
z 1 M , where M max 0 j n 1
aj an
.
The above result has been generalized and improved in various ways. Mohammad [5] used Schwarz Lemma and proved the following result: n
Theorem B. All the zeros of the polynomial
P( z ) a j z j of degree n lie in z j 0
where
M if a n M , an
M max z 1 a n z n 1 ...... a0 max z 1 a0 z n 1 ...... a n1 .
Recently Gulzar [3] proved the following result : n
P( z ) a j z j of degree n lie in the ring-shaped region
Theorem C.All the zeros of the polynomial
j 0
r1 z r2 , where r1
1 3 2
[ R a1 (M a 0 ) 4 a1 R M ] a1 R 2 ( M a 0 ) 4
2
2
2
2
2M 2 2 2M 1
r2
1 3 2
,
,
[ R a n 1 ( M 1 a n ) 4 a n R M ] a n 1 R ( M 1 a n ) 4
2
2
2
M max z R a n z n ...... a1 z , M 1 max z R a0 z n ...... a n 1 z , 1
2