A power mean inequality involving the complete elliptic integrals
type: Item
instance of: scholarly article
title: A power mean inequality involving the complete elliptic integrals
zbMATH Open document ID: 1306.33027
DOI: 10.1216/RMJ-2014-44-5-1661
published in: Rocky Mountain Journal of Mathematics
publication date: 2015-01-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.2755
full work available at URL: https://projecteuclid.org/euclid.rmjm/1420071560
review text: Let \(\mathcal K(r)\) for \(r\in(0,1)\) be a complete elliptic integrals of the first kind \[ \mathcal K(r)=\int_0^{\pi/2}\frac{d\theta} {\sqrt{1-r^2\sin^2\theta}} \] and \(\mathcal K'(r)=\mathcal K(r')\), where \(r'=\sqrt{1-r^2}\). Let \(m(r)\) be a special function defined as \(m(r)=\frac2{\pi}{r'}^2\mathcal K(r)\mathcal K'(r)\) for \(r\in(0,1)\) and \(M_\lambda(x,y)=\left(\frac{x^\lambda+y^\lambda}2\right)^{1/\lambda}\) be the power mean. In this paper one power mean inequality for the special function \(m(r)\) is obtained. The main result consists in the following statement: For \(\lambda\in\mathbb R\) the inequality \[ M_\lambda(m(x),m(y))\leq m(M_\lambda(x,y))\tag{\(\ast\)} \] holds for all \(x,y\in(0,1)\) if and only if \(\lambda\leq0\). The reverse of \((\ast)\) holds for all \(x,y\in(0,1)\) if and only if \(\lambda\geq C>0\), where \(C\) is some constant. Equality holds in \((\ast)\) if and only if \(x=y\). An open problem is formulated: What is the value of \(C\)?
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 6386241
zbMATH Keywords: complete elliptic integrals
zbMATH Keywords: power mean
zbMATH Keywords: inequality
author: Gendi Wang
author: Xiaohui Zhang
author: Yu-Ming Chu
reviewed by: Valery V. Karachik
MaRDI profile type: Publication
arXiv ID: 1302.2755
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OpenAlex ID: W2964073715
description: scientific article