A mean-value inequality for the Gamma function
type: Item
instance of: scholarly article
title: A mean-value inequality for the Gamma function
zbMATH Open document ID: 0970.33001
DOI: 10.1016/S0893-9659(99)00173-1
author: Horst Alzer
published in: Applied Mathematics Letters
publication date: 2001-07-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 26D07
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1428514
zbMATH Keywords: inequalities
zbMATH Keywords: best bounds
zbMATH Keywords: weighted geometric means
zbMATH Keywords: gamma and psi function
reviewed by: János Aczél
MaRDI profile type: Publication
cites work: Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz
cites work: A Stochastic Approach to the Gamma Function
cites work: On some inequalities for the gamma and psi functions
cites work: Mittelwertungleichungen für Lösungen gewisser Differenzengleichungen. (Mean value inequalities for solutions of certain difference equations)
cites work: Q3661894
description: scientific article; zbMATH DE number 1428514