Variants of the Damascus inequality

TL;DR

This paper generalizes the Damascus inequality by characterizing all positive integers m and n for which a specific symmetric inequality holds under a multiplicative constraint, using GA-convexity and Sturm's sequence.

Contribution

It provides a complete characterization of the pairs (m, n) for which the generalized inequality is valid, extending previous results.

Findings

Identified all (m, n) pairs satisfying the inequality.

Analyzed the topological properties of non-solution sets.

Applied GA-convexity and Sturm's sequence in the proof.

Abstract

In 2016, Dannan and Sitnik established the notable Damascus inequality, which features a symmetric structure under a multiplicative constraint. In this study, we consider the natural generalisation of this inequality by characterising all positive integers $m$ and $n$ such that the inequality \[\sum_{j=1}^m\frac{x_j^n-1}{x_{j}^{n+1}+1}\leqslant 0\] holds for any positive real numbers $x_1, \ldots, x_m$ with $\prod_{j=1}^mx_j=1$. Our approach relies on the theories of GA-convexity and Sturm's sequence. For the cases where the inequality fails, we also investigate the topological properties of the set of non-solutions.