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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 684834, 7 pages
http://dx.doi.org/10.1155/2012/684834
Research Article

A Sharp Double Inequality between Seiffert, Arithmetic, and Geometric Means

1College of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China
2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 2 July 2012; Accepted 21 August 2012

Academic Editor: Josef Diblík

Copyright © 2012 Wei-Ming Gong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For fixed 𝑠β‰₯1 and any 𝑑1,𝑑2∈(0,1/2) we prove that the double inequality 𝐺𝑠(𝑑1π‘Ž+(1βˆ’π‘‘1)𝑏,𝑑1𝑏+(1βˆ’π‘‘1)π‘Ž)𝐴1βˆ’π‘ (π‘Ž,𝑏)<𝑃(π‘Ž,𝑏)<𝐺𝑠(𝑑2π‘Ž+(1βˆ’π‘‘2)𝑏,𝑑2𝑏+(1βˆ’π‘‘2)π‘Ž)𝐴1βˆ’π‘ (π‘Ž,𝑏) holds for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘ if and only if 𝑑1βˆšβ‰€(1βˆ’1βˆ’(2/πœ‹)2/𝑠)/2 and 𝑑2√β‰₯(1βˆ’1/3𝑠)/2. Here, 𝑃(π‘Ž,𝑏), 𝐴(π‘Ž,𝑏) and 𝐺(π‘Ž,𝑏) denote the Seiffert, arithmetic, and geometric means of two positive numbers π‘Ž and 𝑏, respectively.