Inequalities for a polynomial and its derivative
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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$$\begin{gathered} M(p',1) \leqslant \frac{n}{2}\{ M(p,1) - m), \hfill \\ M(p,R) \leqslant \left( {\frac{{R^n + 1}}{2}} \right)M(p,1) - m\left( {\frac{{R^n - 1}}{2}} \right),R > 1 \hfill \\ \end{gathered} $$ It is also known that ifp has all its zeros in the closed unit disc, then$$M(p',1) \geqslant \frac{n}{2}\{ M(p,1) - m\} $$. The present paper contains certain generalizations of these inequalities.
V K Jain^{1}
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November 2019
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