Properties of the power-mean and their applications

Abstract

Suppose $w,v>0$, $w\neq v$ and $A_{u}\left (w,v\right) $ is the $u$-order power mean (PM) of $w$ and $v$. In this paper, we completely describe the convexity of $u\mapsto A_{u}\left (w,v\right) $ on $\mathbb{R}$ and $% s\mapsto A_{u\left (s\right) }\left (w,v\right) $ with $u\left (s\right) =\left (\ln 2\right) /\ln \left (1/s\right) $ on $\left (0,\infty \right) $. These yield some new inequalities for PMs, and give an answer to an open problem.

Keywords

• power mean
• power-type mean
• convexity
• inequality