Optimal bounds for Gaussian arithmetic-geometric mean with applications to complete elliptic integral
type: Item
instance of: scholarly article
title: Optimal bounds for Gaussian arithmetic-geometric mean with applications to complete elliptic integral
zbMATH Open document ID: 1347.26053
DOI: 10.1155/2016/3698463
author: Hua Wang
author: Weimao Qian
author: Yu-Ming Chu
published in: Journal of Function Spaces
publication date: 2016-08-11T00:00:00Z
review text: Summary: We present the best possible parameters \(\alpha_1, \beta_1, \alpha_2, \beta_2 \in \mathbb{R}\) and \(\alpha_3, \beta_3 \in(1 / 2,1)\) such that the double inequalities \(Q^{\alpha_1}(a, b) A^{1 - \alpha_1}(a, b) < \mathrm{AG} [A(a, b), Q(a, b)] < Q^{\beta_1}(a, b) A^{1 - \beta_1}(a, b)\), \(\alpha_2 Q(a, b) +(1 - \alpha_2) A(a, b) < \mathrm{AG} [A(a, b), Q(a, b)] < \beta_2 Q(a, b) +(1 - \beta_2) A(a, b)\), \(Q [\alpha_3 a +(1 - \alpha_3) b, \alpha_3 b +(1 - \alpha_3) a] < \mathrm{AG} [A(a, b), Q(a, b)] < Q [\beta_3 a +(1 - \beta_3) b, \beta_3 b +(1 - \beta_3) a]\) hold for all \(a, b > 0\) with \(a \neq b\), where \(A(a, b)\), \(Q(a, b)\), and \(\mathrm{AG}(a, b)\) are the arithmetic, quadratic, and Gauss arithmetic-geometric means of \(a\) and \(b\), respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 6614000
Wikidata QID: Q59126966
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2016/3698463
OpenAlex ID: W2478425286
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description: scientific article; zbMATH DE number 6614000