On the Zeros of Polynomials

Authors : Vk JAIN

Pages : 0-0

Doi:10.1501/Commua1_0000000272

View : 11 | Download : 8

Publication Date : 1977-01-01

Article Type : Research Paper

Abstract :1. Govil and Rahman [1, Theorem 1] have proved the fol- lowing theorem. n Theorem A. Let p insert ignore into journalissuearticles values(z); = £ aj^ k“0 z’^ insert ignore into journalissuearticles values( 0); be a polynomiai of degree n with complex coefficients such that for some a >«-1 I a^ •11-2 a` I »ol- Then p insert ignore into journalissuearticles values(z); has ali its zeros in |z Kj, where Kj is the greatest positive root of the trinomial equation K`+ı - 2K` + 1 = 0. In the same paper [1], they also remark that Theorem A remains true if the polynomial has gaps and non-vanishing coef- ficients , an,’ satisfy a` a,‘n--ı“a I a°2 I O I a I a a”~` I aj We have sharpened the result for the polynomials having gaps and ■we prove Theorem 1. Let p insert ignore into journalissuearticles values(z); z= a„ z” + a„^ z“> a„ Z`2 +... ^2 / 0, be a polynomial of degree n with conıplex coefficients such that “2 1’ “2’- n,j ali being non-nega- tive integers. insert ignore into journalissuearticles values(ii); for some a >0, the coefficients a,j’s satisfy the condition ianl a“-“> a“-”*i2 |a„J.
Keywords : Zeros, Polynomials, Statistics