On Gautschi's harmonic mean inequality for the gamma function
type: Item
instance of: scholarly article
title: On Gautschi's harmonic mean inequality for the gamma function
zbMATH Open document ID: 1021.33002
DOI: 10.1016/S0377-0427(03)00456-4
author: Horst Alzer
published in: Journal of Computational and Applied Mathematics
publication date: 2003-08-25T00:00:00Z
review text: 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 \(n>8\)].
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 1970743
zbMATH Keywords: gamma function
zbMATH Keywords: inequalities
zbMATH Keywords: Gautschi's conjecture
reviewed by: János Aczél
MaRDI profile type: Publication
cites work: Inequalities for the gamma function
cites work: A power mean inequality for the gamma function
cites work: Mean-value inequalities for the polygamma functions
cites work: ON A GAMMA FUNCTION INEQUALITY OF GAUTSCHI
cites work: Inequalities for quasiconformal mappings in space
cites work: A Harmonic Mean Inequality for the Gamma Function
cites work: Some Mean Value Inequalities for the Gamma Function
cites work: Q4527133
full work available at URL: s0377-0427(03)00456-4
OpenAlex ID: W1971705035
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description: scientific article; zbMATH DE number 1970743