Variants of the Damascus inequality

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.