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2012 Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means
Wei-Mao Qian, Zhong-Hua Shen
J. Appl. Math. 2012: 1-14 (2012). DOI: 10.1155/2012/471096

## Abstract

We prove that $\alpha H(a,b)+(1-\alpha )L(a,b)>{M}_{(1-4\alpha )/3}(a,b)$ for $\alpha \in (0,1)$ and all $a,b>0$ with $a\ne b$ if and only if $\alpha \in [1/4,1)$ and $\alpha H(a,b)+(1-\alpha )L(a,b)<{M}_{(1-4\alpha )/3}(a,b)$ if and only if $\alpha \in (0,3\sqrt{345}/80-11/16)$, and the parameter $(1-4\alpha )/3$ is the best possible in either case. Here, $H(a,b)=2ab/(a+b)$, $L(a,b)=(a-b)/(\text{log}\mathrm{ a}-\text{log}\mathrm{ b})$, and ${M}_{p}(a,b)=(({a}^{p}+{b}^{p})/2{)}^{1/p}(p\ne 0)$ and ${M}_{0}(a,b)=\sqrt{ab}$ are the harmonic, logarithmic, and pth power means of a and b, respectively.

## Citation

Wei-Mao Qian. Zhong-Hua Shen. "Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means." J. Appl. Math. 2012 1 - 14, 2012. https://doi.org/10.1155/2012/471096

## Information

Published: 2012
First available in Project Euclid: 14 December 2012

zbMATH: 1235.26014
MathSciNet: MR2889102
Digital Object Identifier: 10.1155/2012/471096