G02034958

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International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISBN: 2319-6491 Volume 2, Issue 3 (February 2013) PP: 49-58

Further Generalizations of Enestrom-Kakeya Theorem M. H. Gulzar Department of Mathematics, University of Kashmir, Srinagar 190006

Abstract:- Many generalizations of the Enestrom –Kakeya Theorem are available in the literature. In this paper we prove some results which further generalize some known results. Mathematics Subject Classification: 30C10,30C15 Keywords and phrases:- Complex number, Polynomial , Zero, Enestrom – Kakeya Theorem.

I.

INTRODUCTION AND STATEMENT OF RESULTS

The Enestrom –Kakeya Theorem (see[6]) is well known in the theory of the distribution of zeros of polynomials and is often stated as follows: Theorem A: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n whose coefficients satisfy

a n  a n 1  ......  a1  a0  0 . Then P(z) has all its zeros in the closed unit disk

z  1.

In the literature there exist several generalizations and extensions of this result. Joyal et al [5] extended it to polynomials with general monotonic coefficients and proved the following result: Theorem B: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n whose coefficients satisfy

a n  a n 1  ......  a1  a 0 . Then P(z) has all its zeros in

z 

a n  a0  a0 an

.

Aziz and zargar [1] generalized the result of Joyal et al [6] as follows: Theorem C: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n such that for some k  1

kan  a n 1  ......  a1  a0 . Then P(z) has all its zeros in

z  k 1 

kan  a0  a0 an

.

For polynomials ,whose coefficients are not necessarily real, Govil and Rahman [2] proved the following generalization of Theorem A: n

Theorem C: If

P ( z )   a j z j is a polynomial of degree n with Re( a j )   j and Im( a j )   j , j 0

j=0,1,2,……,n, such that

 n   n 1  ......   1   0  0 ,

where

 n  0 , then P(z) has all its zeros in 2 n z  1  ( )(   j ) .  n j 0

Govil and Mc-tume [3] proved the following generalisations of Theorems B and C:

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