On a Cyclic Inequality Related to Chebyshev Polynomials

TL;DR

This paper investigates inequalities involving Chebyshev polynomials and cyclic inequalities, establishing bounds and conditions under which these inequalities hold for nonnegative variables, with specific results for certain parameter values.

Contribution

It introduces a new upper bound for weighted geometric means of Chebyshev polynomials and proves a cyclic inequality for specific parameters and dimensions.

Findings

Weighted geometric mean of Chebyshev polynomials is bounded by another Chebyshev polynomial.

The cyclic inequality holds for a=b=1 and n≤8 for all nonnegative variables.

The paper identifies conditions under which the cyclic inequality is valid.

Abstract

We show that any weighted geometric mean of Chebyshev polynomials is bounded from above by another Chebyshev polynomial. We also study a related homogeneous cyclic inequality $$ \left (\sum_{i=1}^n x_i^{(a+b+1)/2} \right )^2 \geq \sum_{i=1}^n x_i \sum_{i=1}^n x_i^a x_{i+1}^b,$$ where $a,b,x_1,\ldots, x_n$ (with $x_{n+1}=x_1$) are nonnegative. In particular, we prove that the inequality holds when $a=b=1$ and $n\leq 8$ for all nonnegative numbers $x_1,\ldots, x_n$.