A Refinement of an Inequality due to Ankeny and Rivilin

Abstract

Let $p(z)= \sum_{\nu =0}^n a_\nu z^\nu$ be a polynomial of degree $n$, $ M(p,R):= \max_{|z|=R \ge 0} |p(z)|,$ and $M(p,1):=M(p)$. Then by well-known result due to Ankeny and Rivlin \cite{Ankeny}, we have
\begin{eqnarray*}
M(p.R)\le \left(\frac{R^n+1}{2} \right) M(p), ~~~R\ge 1.
\end{eqnarray*}
In this paper, we sharpen and generalizes the above inequality by using a result due to Govil \cite{Govil1989}.