On an inequality of Paul Turán

Abstract

Let $P(z)$ be a polynomial and $P^{\prime }(z)$ its derivative. In this paper,we shall obtain certain compact generalizations and sharp refinements of some resultsof Govil, Malik, Turán and others concerning the maximum modulus of $P(z)$ and $P^{\prime }(z)$on the unit circle $|z|=1$, which also yields a number of other interesting results forvarious choices of parameters.