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arXiv · 2609.28944

Sharp power-mean comparisons for Gauss hypergeometric functions

Qintao Deng·Haoran Peng·Jiatong Zou
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Abstract

We determine the sharp weighted power-mean comparisons for the family $\{H_a(r)\}_{r\in (0,1)}$ with the logarithmic interpretation at $a=0$. In the parameter ranges considered by Barnard, Richards and Tiedeman, we give a complete characterization of all orders $λ,μ\in\mathbb{R}$ for which $$A_λ(w;1,1-r)\leq H_a(r)\leq A_μ(w;1,1-r)$$ holds for every $r\in(0,1)$. In particular, our results settle completely their two power-mean conjectures. The two sharp orders are determined by the second-order expansion at $r=0$ and the endpoint matching as $r\to1$. A weighted Wronskian identity and a sign analysis of its residual establish the global inequalities.

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Qintao Deng, Haoran Peng, Jiatong Zou. 2026-09-24. Sharp power-mean comparisons for Gauss hypergeometric functions. https://arxiv.org/abs/2609.28944

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