On sharpening and generalization of Rivlin's inequality

Authors: PRASANNA KUMAR, GRADIMIR MILOVANOVIC

Abstract: n inequality due to T. J. Rivlin from 1960 states that if $P(z)$ is a polynomial of degree $n$ having no zeros in $|z|<1$, then \[ \max_{|z|=r}|P(z)|\geq \left(\frac{1+r}{2}\right)^n\max_{|z|=1}|P(z)|\] for $0\leq r\leq 1$. In this paper, we prove some generalizations of the above Rivlin's inequality which sharpens Rivlin's inequality as a special case. Some important consequences of these results are also discussed and some related inequalities are obtained.

Keywords: Polynomials, zeros, inequalities

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