A sharp double inequality between Seiffert, arithmetic, and geometric means
type: Item
instance of: scholarly article
title: A sharp double inequality between Seiffert, arithmetic, and geometric means
zbMATH Open document ID: 1246.26017
DOI: 10.1155/2012/684834
author: Wei-Ming Gong
author: Ying-Qing Song
author: Miao-Kun Wang
author: Yu-Ming Chu
published in: Abstract and Applied Analysis
publication date: 2012-09-07T00:00:00Z
review text: Summary: For fixed \(s \geq 1\) and any \(t_1, t_2 \in (0, 1/2)\) we prove that the double inequality \(G^s (t_1 a + (1 - t_1)b, t_1 b + (1 - t_1)a) A^{1-s}(a, b) < P(a, b) < G^s(t_2 a + (1 - t_2)b, t_2 b + (1 - t_2)a)A^{1-s}(a, b)\) holds for all \(a, b > 0\) with \(a \neq b\) if and only if \(t_1 \leq (1 - \sqrt{1 - (2/\pi)^{2/s}})/2\) and \(t_2 \geq (1 - 1/\sqrt{3s})/2\). Here, \(P(a, b), A(a, b)\) and \(G(a, b)\) denote the Seiffert, arithmetic, and geometric means of two positive numbers \(a\) and \(b\), respectively.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 6078783
zbMATH Keywords: Seiffert mean
zbMATH Keywords: arithmetic mean
zbMATH Keywords: geometric mean
Wikidata QID: Q58695882
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/684834
OpenAlex ID: W2052471515
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description: scientific article; zbMATH DE number 6078783