On the Erdős–Lax Inequality

Abstract

The Erdős–Lax Theorem states that if $P(z)=\sum _{\nu =1}^n a_{\nu }z^{\nu }$ is a polynomial of degree $n$ having no zeros in $|z|<1,$ then \begin{equation} \max _{|z|=1}|P^{\prime }(z)|\le \frac{n}{2}\max _{|z|=1}|P(z)|. \end{equation} In this paper, we prove a sharpening of the above inequality (1). In order to prove our result we prove a sharpened form of the well-known Theorem of Laguerre on polynomials, which itself could be of independent interest.