On a cyclic inequality related to Chebyshev polynomials

Javaheri, Mohammad; Shen, Harry

doi:10.2140/involve.2024.17.519

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On a cyclic inequality related to Chebyshev polynomials

Mohammad Javaheri and Harry Shen

Vol. 17 (2024), No. 3, 519–530

DOI: 10.2140/involve.2024.17.519

Abstract

We show that any weighted geometric mean of Chebyshev polynomials is bounded from above by another Chebyshev polynomial. We also study the related homogeneous cyclic inequality

{(\sum_{i = 1}^{n} x_{i}^{(a + b + 1) ∕ 2})}^{2} \geq \sum_{i = 1}^{n} x_{i} \sum_{i = 1}^{n} x_{i}^{a} x_{i + 1}^{b},

where $a, b, x_{1}, \dots, x_{n}$ (with $x_{n + 1} = x_{1}$ ) are nonnegative. In particular, we prove that this inequality holds when $a = b = 1$ for all nonnegative numbers $x_{1}, \dots, x_{n}$ if and only if $n \leq 8$ .

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Keywords

Chebyshev polynomials, cyclic homogeneous inequality

Mathematical Subject Classification

Primary: 26D07

Milestones

Received: 28 December 2022

Revised: 30 April 2023

Accepted: 7 May 2023

Published: 17 July 2024

Communicated by Silvestru Sever Dragomir

Authors

Mohammad Javaheri
School of Science Siena College Loudonville, NY United States

Harry Shen
School of Science Siena College Loudonville, NY United States

Vol. 17, No. 3, 2024