## Journal of Applied Mathematics

### An Optimal Double Inequality between Seiffert and Geometric Means

#### Abstract

For $\alpha ,\beta \in (0,1/2)$ we prove that the double inequality $G(\alpha a+(1-\alpha )b,\alpha b+(1-\alpha )a)\lt P(a,b)\lt G(\beta a+(1-\beta )b,\beta b+(1-\beta )a)$ holds for all $a,b>0$ with $a\ne b$ if and only if $\alpha \le (1-\sqrt{1-4/{\pi }^{2}})/2$ and $\beta \ge (3-\sqrt{3})/6$. Here, $G(a,b)$ and $P(a,b)$ denote the geometric and Seiffert means of two positive numbers a and b, respectively.

#### Article information

Source
J. Appl. Math., Volume 2011 (2011), Article ID 261237, 6 pages.

Dates
First available in Project Euclid: 15 March 2012

https://projecteuclid.org/euclid.jam/1331818673

Digital Object Identifier
doi:10.1155/2011/261237

Mathematical Reviews number (MathSciNet)
MR2854959

Zentralblatt MATH identifier
1235.26011

#### Citation

Chu, Yu-Ming; Wang, Miao-Kun; Wang, Zi-Kui. An Optimal Double Inequality between Seiffert and Geometric Means. J. Appl. Math. 2011 (2011), Article ID 261237, 6 pages. doi:10.1155/2011/261237. https://projecteuclid.org/euclid.jam/1331818673