Inequalities for the derivative of a polynomial with restricted zeros

Abstract

For a polynomial p(z) of degree n,  it is known that

$$\begin{aligned}\max _{|z|=1}|p^{'}(z)|\le \frac{n}{1+k}\max _{|z|=1}|p(z)|,\end{aligned}$$

if $p(z)\ne 0$ in $|z|<k,k \ge 1$ and

$$\begin{aligned}\max _{|z|=1}|p^{'}(z)|\ge \frac{n}{1+k}\max _{|z|=1}|p(z)|,\end{aligned}$$

if $p(z)\ne 0$ for $|z|>k,k \le 1.$ In this paper, we assume that there is a zero of multiplicity s,  $s <n$ at a point inside $|z|=1$ and prove some generalizations and improvements of these inequalities.