On Gautschi's harmonic mean inequality for the gamma function (Q1405195)

Proof is offered to \textit{W. Gautschi}'s conjecture [SIAM J. Math. Anal. 5, 282-292 (1974; Zbl 0276.33004)] that \(H_n:=\inf_{(x_1,\dots ,x_n)\in S_n} ((1/n)\sum_{k=1}^n[1/\Gamma(x_k)])^{-1}=1\) for \(n\leq 8,\) where \(S_n=\{(x_1,\dots ,x_n)\in \mathbb{R}^n_+ \mid \prod_{k=1}^n x_k=1\}.\) [In the same paper Gautschi had proved that \(H_n<1\) for \(n>8\)].