Inequalities and bounds for generalized complete elliptic integrals
type: Item
instance of: scholarly article
title: Inequalities and bounds for generalized complete elliptic integrals
zbMATH Open document ID: 1206.33020
author: Edward Neuman
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-10-22T00:00:00Z
review text: The generalized complete elliptic integrals of the first and second kind are defined by \[ {\mathcal K}_{a,b,c}=\frac{B(a,b)}{2} F(a,b;c;k^2), \] and \[ {\mathcal E}_{a,b,c}=\frac{B(a,b)}{2} F(a-1,b;c;k^2), \] where \(B(a,b)=\int_0^1 u^{a-1} (1-u)^{b-1}\,du\) and \(F(a,b;c;z)\) is the Gauss hypergeometric function. The author gives computable bounds for these integrals as well as for their differences and products. Besides other results, he shows that \({\mathcal K}_{a,b,c}\) and \({\mathcal E}_{a,b,c}\) are strictly log-convex functions in \(a\).
reviewed by: Thomas Stoll
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 5804452
zbMATH Keywords: generalized complete elliptic integrals of the first and second kind
zbMATH Keywords: Gauss' hypergeometric function
zbMATH Keywords: power mean
zbMATH Keywords: logarithmic convexity
zbMATH Keywords: concavity/convexity
zbMATH Keywords: inequalities
zbMATH Keywords: lower and upper bounds
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.06.060
OpenAlex ID: W2077374668
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DOI: 10.1016/J.JMAA.2010.06.060
description: scientific article; zbMATH DE number 5804452