## Abstract and Applied Analysis

### The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

#### Abstract

For $p\in \mathbb{R}$, the power mean ${M}_{p}(a,b)$ of order $p$, logarithmic mean $L(a,b)$, and arithmetic mean $A(a,b)$ of two positive real values $a$ and $b$ are defined by ${M}_{p}(a,b)=(({a}^{p}+{b}^{p})/2{)}^{1/p}$, for $p{\,\neq\,}0$ and ${M}_{p}(a,b)=\sqrt{ab}$, for $p=0$, $L(a,b)=(b-a)/(\log b-\log a)$, for $a{\,\neq\,}b$ and $L(a,b)=a$, for $a=b$ and $A(a,b)=(a+b)/2$, respectively. In this paper, we answer the question: for $\alpha \in (0,1)$, what are the greatest value $p$ and the least value $q$, such that the double inequality ${M}_{p}(a,b)\leq \alpha A(a,b)+(1-\alpha )L(a,b)\leq {M}_{q}(a,b)$ holds for all $a,b>0$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 604804, 9 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620716

Digital Object Identifier
doi:10.1155/2010/604804

Mathematical Reviews number (MathSciNet)
MR2629623

Zentralblatt MATH identifier
1190.26038

#### Citation

Xia, Wei-Feng; Chu, Yu-Ming; Wang, Gen-Di. The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means. Abstr. Appl. Anal. 2010 (2010), Article ID 604804, 9 pages. doi:10.1155/2010/604804. https://projecteuclid.org/euclid.aaa/1288620716