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

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N. K. Govil, Eze R. Nwaeze

Complex Variables · Submitted 16 Jan 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1701.04161 (version 1)
Pith IDD3UIQE47, Pith's permanent record of this paper and its review
CitationEze R. Nwaeze, N. K. Govil. "Some Sharpening and Generalizations of a result of T. J. Rivlin." arXiv:1701.04161 (2017). https://pith.science/paper/D3UIQE47
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arXiv listing abstract v1. Results and the main claim are underlined; .

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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$. background 1.00 0.10 0.02 asserted 0.78
s1 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. contribution 0.94 0.93 0.93 asserted 1.00

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Original 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.

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