A mean-value inequality for the Gamma function
The gamma function is geometrically concave or geometrically convex [cf. \textit{L. Lucht}, Aequationes Math. 39, 204-209 (1990; Zbl 0705.35002); \textit{D. Gronau} and \textit{J. Matkowski}, Math. Pannonica 4, No. 2, 153-160 (1993; Zbl 0791.33002)], that is \[ \frac{\prod_{j=1}^n \Gamma(x_j)^{q_j}}{\Gamma(\prod_{j=1}^n x_j^{q_j})}\leq 1 \quad \text{or} \quad \geq 1, \] when \(\sum_{j=1}^n q_j =1\), \(q_j>0,\) and \(x_j\in]0,a]\) or \(x_j\in[a,\infty[\) \((j=1,\dots,n),\) respectively. Here \(x=a\) is the unique positive solution of the equation \(\Gamma(x+1)[\Gamma''(x+1)+(1/x)\Gamma'(x+1)]=\Gamma'(x+1)^2.\) In the second case there is no upper bound for the above quotient. For the first case the author offers \(\Gamma(a+1)^{1-\min_{: 1\leq j\leq n}: q_j}\) as best possible lower bound.
scientific article; zbMATH DE number 1428514