On the Zeros of Polynomials

Year 1977, Volume: 26 , 0 - 0, 01.01.1977

V.k. Jaın

https://doi.org/10.1501/Commua1_0000000272

Abstract

1. Govil and Rahman [1, Theorem 1] have proved the fol- lowing theorem.
n Theorem A. Let p (z) = £ aj^ k“0 z’^ ( 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 (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 (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.
(ii) for some a >0, the coefficients a,j’s satisfy the condition ianl a“-“> a“-”*i2 |a„J.

Keywords

Zeros, Polynomials, Statistics

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi