An improved inequality for \(k\)-th derivative of a polynomial (Q2709921)

In this paper the author proves the following result: Theorem 1.1. Let \(p(z)\) be a polynomial of degree \(n\). Then NEWLINE\[NEWLINE\begin{multlined} \left|p^{(k)} (\beta)\right|\leq \\ \leq {n (n-1) \dots(n-k+1) \over|\beta|^k} \left[{1\over 2^k}\left\{ \left|p(\beta) \right|+ \max_{1\leq i\leq n} \left|p(\beta z_i) \right|\right\} + \left( 1-{1\over 2^{k-1}} \right) \max_{1\leq t\leq 2n} \left|p(\beta b_t) \right|\right],\end{multlined}NEWLINE\]NEWLINE \(\beta\neq 0\) and \(k\geq 1\), where \(z_1,z_2, \dots,z_n\) are the zeros of \(z^n+1\) and \(b_1,b_2, \dots,b_{2n}\) are the zeros of \(z^{2n}-1\). The inequality is sharp and the extremal polynomial is \(p(z)=\alpha z^n\).