SOME GENERALIZATIONS OF A THEOREM OF PAUL TURÀN CONCERNING POLYNOMIALS

Abstract

Let $P(z)=\sum _{\nu =0}^n{a}_{\nu }{z}^{\nu }$ be a polynomial of degree $n$ having all its zeros in $|z|\le k$, $k\ge 1$. It was shown by Govil that$\underset{|z|=1}{max}|P^{\prime }(z)|\ge \frac{n}{1+k^n}\underset{|z|=1}{max}|P(z)|$.In this paper, we shall obtain some sharp estimates by involving the coefficients which not only refine the above result but also generalise some well-known results of this type.