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Some Sharpening and Generalizations of a result of T. J. Rivlin by Govil, N. K.; Nwaeze, Eze R. is a scholarly article available to read on EtoBox.

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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.

Author

Govil, N. K.; Nwaeze, Eze R.

Published

2017

Language

EN