Some Sharpening and Generalizations of a result of T. J. Rivlin

Abstract

Let $p(z)=a_0+a_{1}z+a_2{z}^2+a_3{z}^3+\cdots +a_n{z}^n$ be a polynomial of degree $n$. Rivlin \cite{Rivlin} proved that if $p(z)\ne 0$ in the unit disk, then for $0<r\le 1$, $\underset{|z|=r}{\max }|p(z)|\ge (\frac{r+1}{2}{)}^n\underset{|z|=1}{\max }|p(z)|.$ ~In this paper, we prove a sharpening and generalization of this result, and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin's Theorem.