On a Conjecture by Ren and Li

2025/09/26 by Luo, Yongbing, Yan, Ping
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.21757

Abstract

This paper proves a conjecture proposed by Ren and Li (2015: 393, 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 ≥ 1 and all x > 0. Crucially, we establish the inequality S_m(x) > σ_m(x) unconditionally, requiring no parity conditions, size conditions on m, or lower bound on x.

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