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Sharp power-mean comparisons for Gauss hypergeometric functions
Abstract
We determine the sharp weighted power-mean comparisons for the family {Ha(r)}r∈ (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∈ℝ for which Aλ(w;1,1−r)≤ Ha(r)≤ Aμ(w;1,1−r) holds for every r∈(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→1. A weighted Wronskian identity and a sign analysis of its residual establish the global inequalities.
Related
- Strict Total Positivity of a Gauss Hypergeometric Kernel, and the Sharp Threshold for the One-Dimensional SL(2,ℝ) Conformal Block
- Combinatorial identities derived from explicit formulas of Gauss hypergeometric functions
- A Power Geometry and Hypergeometric Functions
- The second type of power mean bounds for the hypergeometric mean
- The first type of power mean bounds for the Gaussian hypergeometric function
BibTeX
@misc{deng2026sharp,
title = {{Sharp power-mean comparisons for Gauss hypergeometric functions}},
author = {Qintao Deng and Haoran Peng and Jiatong Zou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28944},
eprint = {2609.28944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28944},
}
RIS
TY - UNPB
TI - Sharp power-mean comparisons for Gauss hypergeometric functions
AU - Qintao Deng
AU - Haoran Peng
AU - Jiatong Zou
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28944
UR - https://arxiv.org/abs/2609.28944
AB - We determine the sharp weighted power-mean comparisons for the family {H_a(r)}_r∈ (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∈ℝ for which A_λ(w;1,1−r)≤ H_a(r)≤ A_μ(w;1,1−r) holds for every r∈(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→1. A weighted Wronskian identity and a sign analysis of its residual establish the global inequalities.
ER -