On Ivády's bounds for the gamma function and related results
type: Item
instance of: scholarly article
title: On Ivády's bounds for the gamma function and related results
zbMATH Open document ID: 1499.33005
author: Horst Alzer
author: Man Kam Kwong
published in: Periodica Mathematica Hungarica
publication date: 2021-08-03T00:00:00Z
review text: Improving a bound given by \textit{P. Ivády} [J. Math. Inequal. 3, No. 2, Article ID 23, 227--236 (2009; Zbl 1179.26052)] and answering a question of \textit{Z.-H. Yang} et al. [J. Inequal. Appl. 2017, Paper No. 210, 17 p. (2017; Zbl 1370.41056)], the authors prove that the inequality \[ \Gamma(x+1)\leq \frac{x^2+\beta}{x+\beta} \] holds for all \(x\in [0,1]\), \(\beta\geq \beta^*\), with the best possible constant \[\beta^{*}=\max_{0.1\leq x\leq 0.3}f(x)= 1.75527\ldots,\] where \(f\) is given by \[ f(x)=\frac{x\Gamma(x+1)-x^2}{1-\Gamma(x+1)}. \] Moreover, they show that \(f\) is strictly concave on \([0, 1]\) and apply this result to obtain some interesting functional inequalities for the gamma function.
reviewed by: Ioan Raşa
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 7377556
zbMATH Keywords: gamma function
zbMATH Keywords: inequalities
zbMATH Keywords: concavity
zbMATH Keywords: optimal constant
MaRDI profile type: Publication
full work available at URL: s10998-020-00356-9
OpenAlex ID: W3044996896
cites work: Q5558293
cites work: Inequalities for Euler's gamma function
cites work: Gamma function inequalities
cites work: Revisiting some properties of the gamma function
cites work: Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz
cites work: A Sharpening of Wielandt's Characterization of the Gamma Function
cites work: A Stochastic Approach to the Gamma Function
cites work: Q3391720
cites work: Q3473576
cites work: Gamma triads
cites work: Wielandt's Theorem About the Γ-Function
cites work: The Gamma Function: An Eclectic Tour
cites work: Q5625295
cites work: EXPRESSIONS FOR VALUES OF THE GAMMA FUNCTION
cites work: On rational bounds for the gamma function
DOI: 10.1007/S10998-020-00356-9
description: scientific article; zbMATH DE number 7377556