Some Sharpening and Generalizations of a result of T. J. Rivlin

TL;DR

This paper improves upon Rivlin's classical inequality for polynomials non-vanishing in the unit disk by providing sharper bounds and generalizations, supported by examples demonstrating significant improvements.

Contribution

The authors present a sharpened and generalized version of Rivlin's inequality, enhancing the bounds for polynomial maximum modulus in the unit disk.

Findings

New bounds that improve Rivlin's inequality for certain polynomials

Generalizations that extend the applicability of the original result

Examples illustrating the effectiveness of the new bounds

Abstract

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$. Rivlin \cite{Rivlin} proved that if $p(z)\neq 0$ in the unit disk, then for $0<r\leq 1$, $\displaystyle{\max_{|z| = r}|p(z)|} \geq \Big(\dfrac{r+1}{2}\Big)^n \displaystyle{\max_{|z|=1} |p(z)|}.$ ~In this paper, we prove a sharpening and generalization of this result, and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin's Theorem.