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 <Equation ID="Equ9"> <EquationSource Format="TEX">$\begin{aligned} \sum _{j=1}^m\frac{x_j^n-1}{x_{j}^{n+1}+1}\leqslant 0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mfrac> <mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mi>n</mi> </msubsup> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>⩽</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>holds for any positive real numbers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$x_1, \ldots , x_m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>m</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\prod _{j=1}^mx_j=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∏</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.

Variants of the Damascus inequality

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2026年2月21日

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Chanatip Sujsuntinukul,
Christophe Chesneau

摘要

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 $\begin{aligned} \sum _{j=1}^m\frac{x_j^n-1}{x_{j}^{n+1}+1}\leqslant 0 \end{aligned}$ $\begin{matrix} \sum_{j = 1}^{m} \frac{x_{j}^{n} - 1}{x_{j}^{n + 1} + 1} ⩽ 0 \end{matrix}$ holds for any positive real numbers $x_1, \ldots , x_m$ $x_{1}, \dots, x_{m}$ with $\prod _{j=1}^mx_j=1$ $\prod_{j = 1}^{m} x_{j} = 1$ . 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.