# Optimal inequalities for the power, harmonic and logarithmic means

Document Type : Research Paper

Authors

1 Huzhou Teachers College

2 Hebei University

3 Hunan University

Abstract

For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers $a$ and $b$, respectively.

Keywords

### History

• Receive Date: 16 October 2009
• Revise Date: 12 March 2011
• Accept Date: 12 March 2011