Inequalities for the derivative of polynomials with restricted zeros

Abstract

For a polynomial $\mathit{P(z)=\sum_{\nu =0}^{n}a_{\nu}z^{\nu}}$ of degree $\mathit{n}$ having all its zeros in $\mathit{|z|\leq k,k \geq 1}$, it was shown by Rather and Dar \cite{1} that $$\max_{|z|=1} |P^{\prime}(z)|\geq \frac{1}{1+k^n}\bigg(n+\frac{k^n|a_n|-|a_0|}{k^n|a_n|+|a_0|}\bigg)\max_{|z|=1}|P(z)|.$$ In this paper, we shall obtain some sharp estimates, which not only refine the above inequality but also generalize some well known Tur\'{a}n-type inequalities.