## Abstract and Applied Analysis

### Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means

#### Abstract

We prove that the double inequalities ${I}^{{\alpha }_{1}}\left(a,b\right){Q}^{1-{\alpha }_{1}}\left(a,b\right) hold for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_{1}\ge 1/2$, ${\beta }_{1}\le \mathrm{log}\left[\sqrt{2}\mathrm{log}\left(1+\sqrt{2}\right)\right]/\left(1-\mathrm{log}\sqrt{2}\right)$, ${\alpha }_{2}\ge 5/7$, and ${\beta }_{2}\le \mathrm{log}\left[2\mathrm{log}\left(1+\sqrt{2}\right)\right]$, where $I\left(a,b\right)$, $M\left(a,b\right)$, $Q\left(a,b\right)$, and $C\left(a,b\right)$ are the identric, Neuman-Sándor, quadratic, and contraharmonic means of $a$ and $b$, respectively.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 348326, 12 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511798

Digital Object Identifier
doi:10.1155/2013/348326

Mathematical Reviews number (MathSciNet)
MR3035385

Zentralblatt MATH identifier
1276.26065

#### Citation

Zhao, Tie-Hong; Chu, Yu-Ming; Jiang, Yun-Liang; Li, Yong-Min. Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means. Abstr. Appl. Anal. 2013 (2013), Article ID 348326, 12 pages. doi:10.1155/2013/348326. https://projecteuclid.org/euclid.aaa/1393511798

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