Year: 2017
Analysis in Theory and Applications, Vol. 33 (2017), Iss. 3 : pp. 219–228
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 [12] proved that if $p(z)\neq 0$ in the unit disk, then for $0< r\leq 1,$ $${\max_{|z|=r}|p(z)|}\geq \Big(\dfrac{r+1}{2}\Big)^n{\max_{|z|=1}|p(z)|}.$$ 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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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2017.v33.n3.3
Analysis in Theory and Applications, Vol. 33 (2017), Iss. 3 : pp. 219–228
Published online: 2017-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 10
Keywords: Inequalities polynomials zeros.
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Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomial
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Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomial
Different types of Bernstein inequalities
Gardner, Robert B. | Govil, Narendra K. | Milovanović, Gradimir V. | Rassias, Themistocles M.2022
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Improvement and generalization of polynomial inequality of T. J. Rivlin
Devi, Maisnam Triveni | Singh, Thangjam Birkramjit | Chanam, BarchandSão Paulo Journal of Mathematical Sciences, Vol. 17 (2023), Iss. 2 P.1006
https://doi.org/10.1007/s40863-022-00300-4 [Citations: 0]