Sharp power-mean comparisons for Gauss hypergeometric functions

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 $\lambda ,\mu \in \mathbb{R}$ for which $$A_{\lambda }(w;1,1-r)\le H_a(r)\le A_{\mu }(w;1,1-r)$$ 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 $r\to 1$. A weighted Wronskian identity and a sign analysis of its residual establish the global inequalities.