Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
type: Item
instance of: scholarly article
title: Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
zbMATH Open document ID: 1485.26058
author: Hong-Hu Chu
author: Tiehong Zhao
author: Yu-Ming Chu
published in: Mathematica Slovaca
publication date: 2021-12-07T00:00:00Z
review text: Let \(n\in \mathbb{Z} \) and \(a,b>0.\) Then the Toader mean \(T_{n}(a,b)\) of order \(n\) is defined by \[ T_{n}(a,b)=\left\{\begin{array}{cc} \left( \frac{2}{\pi}\int\nolimits_{0}^{\frac{\pi }{2}}\sqrt{a^{n}\cos ^{2}\theta +b^{n}\sin ^{2}\theta }d\theta \right) ^{\frac{2}{n}}, & n\neq 0 \\ \sqrt{ab} & n=0. \end{array} \right. \] In this paper, the authors study the bounds for the Toader mean of order \(3 \) in terms of the arithmetic, quadratic and contraharmonic means. They derive and prove in Theorems 3.1, 3.2 and 3.3 the best possible parameters \(\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}\in \mathbb{R}\) and \(\alpha _{3},\beta _{3}\in \lbrack \frac{1}{2},1]\) for which the double inequalities \[ \alpha _{1}C(a,b)+(1-\alpha _{1})A(a,b)<T_{3}(a,b)<\beta _{1}C(a,b)+(1-\beta _{1})A(a,b), \] \[ \alpha _{2}C(a,b)+(1-\alpha _{2})Q(a,b)<T_{3}(a,b)<\beta _{2}C(a,b)+(1-\beta _{2})Q(a,b), \] and \[ C(\alpha _{3};a;b)<T_{3}(a;b)<C(\alpha _{3};a;b) \] hold for \(a, b>0\) with \(a\neq b.\) Furthermore, new sharp bounds for the complete elliptic integral of the second kind were provided and proved, where \(A(a;b)=\frac{(a+b)}{2}\) is the arithmetic mean, \(Q(a;b)=\sqrt{\frac{ (a^{2}+b^{2})}{2}}\) is the quadratic mean, \(C(a;b)=\frac{(a^{2}+b^{2})}{(a+b) }\) is the contra-harmonic mean, \(C(p;a,b)=C[pa+(1-p)b;pb+(1-p)a]\) is the one-parameter contra-harmonic mean and \(T_{3}(a;b)=\left( \frac{2}{\pi } \int\nolimits_{0}^{\frac{\pi }{2}}\sqrt{a^{3}\cos ^{2}\theta +b^{3}\sin ^{2}\theta }d\theta \right) ^{\frac{2}{3}}\) is the Toader mean of order \(3.\)
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 7439831
zbMATH Keywords: Toader mean
zbMATH Keywords: complete elliptic integral
zbMATH Keywords: arithmetic mean
zbMATH Keywords: quadratic mean
zbMATH Keywords: contraharmonic mean
MaRDI profile type: Publication
full work available at URL: ms-2017-0417
OpenAlex ID: W3090884723
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DOI: 10.1515/MS-2017-0417