Abstract and Applied Analysis

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

Wei-Ming Gong, Ying-Qing Song, Miao-Kun Wang, and Yu-Ming Chu

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Abstract

For fixed s 1 and any t 1 , t 2 ( 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 b if and only if t 1 ( 1 - 1 - ( 2 / π ) 2 / s ) / 2 and t 2 ( 1 - 1 / 3 s ) / 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.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 684834, 7 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495846

Digital Object Identifier
doi:10.1155/2012/684834

Mathematical Reviews number (MathSciNet)
MR2965473

Zentralblatt MATH identifier
1246.26017

Citation

Gong, Wei-Ming; Song, Ying-Qing; Wang, Miao-Kun; Chu, Yu-Ming. A Sharp Double Inequality between Seiffert, Arithmetic, and Geometric Means. Abstr. Appl. Anal. 2012 (2012), Article ID 684834, 7 pages. doi:10.1155/2012/684834. https://projecteuclid.org/euclid.aaa/1355495846


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