Sharp inequalities involving the power mean and complete elliptic integral of the first kind
type: Item
instance of: scholarly article
title: Sharp inequalities involving the power mean and complete elliptic integral of the first kind
zbMATH Open document ID: 1314.33019
DOI: 10.1216/RMJ-2013-43-5-1489
author: Yu-Ming Chu
author: Song-Liang Qiu
author: Miao-Kun Wang
published in: Rocky Mountain Journal of Mathematics
publication date: 2013-11-13T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1382705664
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 6226518
zbMATH Keywords: complete elliptic integrals
zbMATH Keywords: power mean
zbMATH Keywords: inequality
MaRDI profile type: Publication
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OpenAlex ID: W2053344326
review text: For \(0<r<1\), define the complete elliptic integral of the first kind by NEWLINE\[NEWLINEK(r)=\int_{0}^{\pi/2} \frac{d\theta}{\sqrt{1 - r^{2} \sin^{2} \theta}}.NEWLINE\]NEWLINE NEWLINEFor each real number \(p\), denote the power mean of order \(p\) of two positive numbers \(x\) and \(y\) by NEWLINENEWLINE\[NEWLINEM_{p}(x,y) = (x^{p} + y^{p}/2)^{1/p}\quad \text{for }\;p \neq 0NEWLINE\]NEWLINE NEWLINEand NEWLINENEWLINE\[NEWLINEM_{p}(x,y)=\sqrt{xy}\quad \text{for }\;p =0.NEWLINE\]NEWLINE NEWLINEThe main results of this paper are the inequalities NEWLINE\[NEWLINEM_{p}(K(r), K(r')) \geq K(\sqrt{2}/2)NEWLINE\]NEWLINE and NEWLINE\[NEWLINEM_{q}(K(r), K(r'))\leq K(\sqrt{2}/2).NEWLINE\]NEWLINE These are shown to hold for all \(r\in (0,1)\) and \(r' = \sqrt{1 - r^{2}}\) if and only if NEWLINE\[NEWLINEp\geq \frac{1-4[K(\sqrt{2}/2)]^4}{\pi^{2}}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEq\leq \frac{\log 2}{\log(\pi /2) - \log K(\sqrt{2}/2)}.NEWLINE\]
description: scientific article; zbMATH DE number 6226518