An inequality for Tchebycheff polynomials and extensions

Richard Askey; George Gasper; Lawrence A. Harris

doi:10.1016/0021-9045(75)90061-1

An inequality for Tchebycheff polynomials and extensions

Richard Askey, George Gasper, Lawrence A. Harris

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The inequality T_n(xy) ≤ T_n(x) T_n(y), x, y ≥ 1, where T_n(x) is the Tchebycheff polynomial of the first kind, can be proven very easily by use of one of the extremal properties of these polynomials. It also follows from ( d² du²) log T_n(e^u) ≤ 0, u ≥ 0. Various proofs are given for these inequalities and for generalizations to other classes of polynomials.

Original language	English
Pages (from-to)	1-11
Number of pages	11
Journal	Journal of Approximation Theory
Volume	14
Issue number	1
DOIs	https://doi.org/10.1016/0021-9045(75)90061-1
State	Published - May 1975

Bibliographical note

Funding Information:
* Research supported in part by NSF Grant GP-33897. + Research supported in part by NSF Grant GP-32116 and in part by the Alfred Foundation. * Research supported in part by NSF Grant GP-33117.

ASJC Scopus subject areas

Analysis
Numerical Analysis
General Mathematics
Applied Mathematics

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10.1016/0021-9045(75)90061-1

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title = "An inequality for Tchebycheff polynomials and extensions",

abstract = "The inequality Tn(xy) ≤ Tn(x) Tn(y), x, y ≥ 1, where Tn(x) is the Tchebycheff polynomial of the first kind, can be proven very easily by use of one of the extremal properties of these polynomials. It also follows from ( d2 du2) log Tn(eu) ≤ 0, u ≥ 0. Various proofs are given for these inequalities and for generalizations to other classes of polynomials.",

author = "Richard Askey and George Gasper and Harris, {Lawrence A.}",

note = "Funding Information: * Research supported in part by NSF Grant GP-33897. + Research supported in part by NSF Grant GP-32116 and in part by the Alfred Foundation. * Research supported in part by NSF Grant GP-33117.",

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AU - Harris, Lawrence A.

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Y1 - 1975/5

N2 - The inequality Tn(xy) ≤ Tn(x) Tn(y), x, y ≥ 1, where Tn(x) is the Tchebycheff polynomial of the first kind, can be proven very easily by use of one of the extremal properties of these polynomials. It also follows from ( d2 du2) log Tn(eu) ≤ 0, u ≥ 0. Various proofs are given for these inequalities and for generalizations to other classes of polynomials.

AB - The inequality Tn(xy) ≤ Tn(x) Tn(y), x, y ≥ 1, where Tn(x) is the Tchebycheff polynomial of the first kind, can be proven very easily by use of one of the extremal properties of these polynomials. It also follows from ( d2 du2) log Tn(eu) ≤ 0, u ≥ 0. Various proofs are given for these inequalities and for generalizations to other classes of polynomials.

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An inequality for Tchebycheff polynomials and extensions

Abstract

Bibliographical note

ASJC Scopus subject areas

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