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    • Inequalities for a polynomial and its derivative

      V K Jain

      Volume 110 Issue 2 May 2000 pp 137-146

    • Fulltext

       

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/110/02/0137-0146

    • Keywords

       

      Inequalities; zeros; polynomial

    • 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.

    • Author Affiliations

       

      V K Jain1

      1. Mathematics Department, Indian Institute of Technology, Kharagpur - 721 302, India

    • Dates

       
      Published
      May 2000
      Manuscript received
      29 July 1999
      Manuscript revised
      20 December 1999

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