Inequalities for a polynomial and its derivative

For an arbitrary entire functionf and anyr>0, letM(f,r):=max|z|=r |f(z)|. It is known that ifp is a polynomial of degreen having no zeros in the open unit disc, andm:=min|z|=1|p(z)|, then\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{gathered} M(p',1) \leqslant \frac{n}{2}\{ M(p,1) - m), \hfill \\ M(p,R) \leqslant \left( {\frac{{R^n + 1}}{2}} \right)M(p,1) - m\left( {\frac{{R^n - 1}}{2}} \right),R > 1 \hfill \\ \end{gathered} $$ \end{document} It is also known that ifp has all its zeros in the closed unit disc, then\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$M(p',1) \geqslant \frac{n}{2}\{ M(p,1) - m\} $$ \end{document}. The present paper contains certain generalizations of these inequalities.