On the location of the zeros of certain polynomials

Abstract

We extend Aziz and Mohammad’s result that the zeros, of a polynomial P(z) =Σn j=0 ajzj, taj ≥ aj−1 > 0, j=2,3,...,n for certain t(>0), with moduli greater than t(n−1)/n are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial P(z), of degree n, with complex coefficients, does not vanish in the disc |z−aeiα| 0, max |z|=a |P(z)| = |P(aeiα)|, for r < a < 2,r being the greatest positive root of the equation xn−2xn−1+1=0, and finally obtained an upper bound, for moduli of all zeros of a polynomial,(better, in many cases, than those obtainable from many other known results).