Variants of the Damascus inequality
Chanatip Sujsuntinukul, Christophe Chesneau
TL;DR
The paper resolves the generalized Damascus inequality by characterizing all pairs of positive integers (m,n) for which S_n^m\le 0 on the multiplicative constraint surface H_m. It employs GA-convexity and Sturm's sequence to establish sharp bounds in low-dimensional cases and to identify when counterexamples arise, delivering a complete table of valid pairs: (m,n)=(1, any n), (2, any n), (3 with n\in\{1,2,3\}), and (4,1), (5,1) as the only higher-m exceptions, while all other cases with n\ge 2 (and (6,1)) fail. For m=3, the inequality holds for n=1,2 and n=3, with n\ge 4 giving explicit violations; the case n=3 additionally yields a structured proof via GA-convexity, with (1,1,1) as a strict local maximum. The work also analyzes the topology of non-solution sets, proving boundedness and separation from coordinate planes for m=3, and outlines several open questions for larger m regarding the geometry and monotonicity of violation sets. Overall, the results clarify the exponent range under multiplicative constraint structures and reveal intricate connections between convexity notions and polynomial-exponential analysis in inequality theory.
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.