On a Conjecture by Ren and Li

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 $m\ge 1$ 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 $x$.