# Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means

Document Type : Research Paper

Authors

Huzhou Teachers College

Abstract

We find the greatest values $alpha_{1}$ and $alpha_{2}$, and the least values $beta_{1}$ and $beta_{2}$ such that the inequalities $alpha_{1} C(a,b)+(1-alpha_{1} )H(a,b)<L(a,b)<beta_{1} C(a,b)+(1-beta_{1} )H(a,b)$ and $alpha_{2} C(a,b)+(1-alpha_{2}) H(a,b)<I(a,b)<beta_{2} C(a,b)+(1-beta_{2} )H(a,b)$ hold for all $a,b>0$ with $aneq b$. Here, $C(a,b)$, $H(a,b)$, $L(a,b)$, and $I(a,b)$ are the centroidal, harmonic, logarithmic, and identric means of two positive numbers $a$ and $b$, respectively.

Keywords

### History

• Receive Date: 11 June 2011
• Revise Date: 15 January 2012
• Accept Date: 15 January 2012