## Abstract and Applied Analysis

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

#### Abstract

For fixed $s\ge 1$ 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) holds for all $a,b>0$ with $a\ne b$ if and only if ${t}_{1}\le (1-\sqrt{1-(2/\pi {)}^{2/s}})/2$ and ${t}_{2}\ge (1-1/\sqrt{3s})/2$. 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.

#### Article information

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

Dates
First available in Project Euclid: 14 December 2012

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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