## Abstract and Applied Analysis

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

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

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

#### Abstract

For fixed $s\ge 1$ and any ${t}_{1},{t}_{2}\in (\mathrm{0,1}/2)$ we prove that the double inequality $$ 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

**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