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Inequalities for a polynomial and its derivative

Authors:

V K Jain

Institution:

(1) Mathematics Department, Indian Institute of Technology, 721 302 Kharagpur, India

Abstract:

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.

Keywords:

Inequalities zeros polynomial

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