A Ring-Shaped Region Containing All or A Specific Number of The Zeros of A Polynomial

Page 1

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 n1 .

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


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