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arXiv · 2509.21757

On a Conjecture by Ren and Li

Yongbing Luo·Ping Yan
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Abstract

This paper proves a conjecture proposed by Ren and Li (2015: 393, \emph{Journal of Inequalities and Applications}). Our result eliminates the constraints on the parity and size of $m$, as well as the restriction $x > 1$, required in Ren and Li's theorem. Consequently, it fully subsumes their results while extending validity to all integers $m \geq 1$ and all $x > 0$. Crucially, we establish the inequality $S_m(x) > \sigma_m(x)$ unconditionally, requiring no parity conditions, size conditions on $m$, or lower bound on $x$.

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Yongbing Luo, Ping Yan. 2025-09-26. On a Conjecture by Ren and Li. https://arxiv.org/abs/2509.21757

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