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 $|z|\leq k$, $ k\geq 1$. It was shown by Govil that$\underset{|z|=1}\max|P'(z)|\geq\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.