I1057581

Page 1

International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 10, Issue 5 (May 2014), PP.75-81

On Some Extensions of Enestrom-Kakeya Theorem M. H. Gulzar Department of Mathematics University of Kashmir, Srinagar 19000 Abstract: In this paper we generalize some recent extensions of the Enestrom-Kakeya Theorem concerning the location of zeros of polynomials. Mathematics Subject Classification: 30 C 10, 30 C 15 Keywords and Phrases: Coefficient, Polynomial, Zero

I.

INTRODUCTION AND STATEMENT OF RESULTS

The following result known as the Enestrom-Kakeya Theorem [7] is an elegant result in the theory of distribution of the zeros of a polynomial: Theorem A: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n such that

a n  a n 1  ......  a1  a 0  0 . Then all the zeros of P(z) lie in the closed disk

z  1.

In the literature [1-8], there exist several extensions and generalizations of Theorem A. Joyal, Labelle and Rahman [5] gave the following generalization of Theorem A: n

Theorem B: Let

P ( z )   a j z j be a polynomial of degree n such that j 0

a n  a n 1  ......  a1  a 0 . Then all the zeros of P(z) lie in the closed disk z 

an  a0  a0 an

.

Aziz and Zargar [1] generalized Theorems A and B by proving the following results: 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 all the zeros of P(z) lie in the closed disk

z  k 1 

kan  a0  a0 an

.

Recently, Zargar [8] gave the following generalizations of the above mentioned results: n

j Theorem D: Let P ( z )   a j z be a polynomial of degree n  2 such that for some k  1, either j 0

kan  a n  2  .......  a 3  a1  0 and a n 1  a n 3  .......  a 2  a 0  0 , if n is odd or

kan  a n  2  .......  a 2  a 0  0 and a n 1  a n 3  .......  a 3  a1  0 , if n is even. Then all the zeros of P(z) lie in the region

z  z    k 

a n 1 , an 75


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