Rate of growth of polynomials non vanishing inside a circle

Abstract

For a polynomial $P(z)=\displaystyle\sum_{v=0}^na_vz^v$ of degree $n$ having all zeros in $|z|\geq k, k\geq 1$ Govil et al.[\emph{ILLINOIS J. of Math.}] proved: $$|P'(z)|\leq n\dfrac{n|a_0|+k^2|a_1|}{(1+k^2)n|a_0|+2k^2|a_1|}|P(z)|.$$ In this paper besides the refinement of above inequality, we also generalize some well known inequalities.