Some inequalities for the Chi square distribution function and the exponential functions
type: Item
instance of: scholarly article
title: Some inequalities for the Chi square distribution function and the exponential functions
zbMATH Open document ID: 0778.33001
DOI: 10.1007/BF01202311
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: For \(u,x>0\) let \(\gamma(u)\) be the gamma function and let \(\gamma(u,x)=\int_ 0^ x e^{-t} t^{u-1} dt\) (incomplete Gamma function). Let \(P(u,x)=\gamma(u,x)/\Gamma(u)\). We prove several inequalities related to the function \(P\). Main ones are the following: \[ \begin{aligned} {{P(u+\beta,x)} \over {P^{1-\beta}(u,x)P^ \beta(u+1,x)}} &< (u+1)^ \beta {{\Gamma(u+1)} \over {\Gamma(u+\beta+1)}}, \qquad (\beta\in(0,1)),\\ {{P^{\beta-1}(u,x)P(u+\beta,x)} \over {P^ \beta(u+1,x)}} &> (u+1)^ \beta {{\Gamma(u+1)} \over {\Gamma(u+\beta+1)}}, \qquad (\beta\in(-u,0)\cup (1,+\infty)),\\ {{P(u,x)\Gamma(u+2,x)} \over {P^ 2(u+1,x)}} &< 1.\end{aligned} \] The first inequality is an improvement of an author's inequality [Publ. Elektrothehn. Fak., Univ. Beogr. Ser. Mat. 2, 89-94 (1991; Zbl 0748.60021)]. The second inequality for \(\beta=2\) and \(u\in \mathbb{N}\) reduces to an inequality for residual after the \(n\)th term of the Maclaurin expansion for \(e^ x\) [\textit{H. Alzer}, Arch. Math. 55, 462-464 (1990; Zbl 0723.26007)]. The third inequality gives rise to a conjecture that the function \(u\mapsto P(u,x)\) is log-concave.
Mathematics Subject Classification ID: 33B10
Mathematics Subject Classification ID: 26D07
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 446374
zbMATH Keywords: convexity
zbMATH Keywords: Gamma function
zbMATH Keywords: exponential function
author: Milan Merkle
MaRDI profile type: Publication
cites work: An inequality for the exponential function
cites work: Q3990208
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cites work: Some theorems on residues
cites work: Q5520081
description: scientific article; zbMATH DE number 446374