## Abstract and Applied Analysis

### Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means

#### Abstract

We answer the question: for $\alpha ,\beta ,\gamma \in (0,1)$ with $\alpha +\beta +\gamma =1$, what are the greatest value $p$ and the least value $q$, such that the double inequality ${L}_{p}(a,b)\lt{A}^{\alpha }(a,b){G}^{\beta }(a,b){H}^{\gamma }(a,b)\lt{L}_{q}(a,b)$ holds for all $a,b\gt0$ with $a{\,\neq\,}b$? Here ${L}_{p}(a,b)$, $A(a,b)$, $G(a,b)$, and $H(a,b)$ denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbers $a$ and $b$, respectively.

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 303286, 13 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620708

Digital Object Identifier
doi:10.1155/2010/303286

Mathematical Reviews number (MathSciNet)
MR2607125

Zentralblatt MATH identifier
1185.26064

#### Citation

Chu, Yu-Ming; Long, Bo-Yong. Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means. Abstr. Appl. Anal. 2010 (2010), Article ID 303286, 13 pages. doi:10.1155/2010/303286. https://projecteuclid.org/euclid.aaa/1288620708