Optimal power mean bounds for the second Yang mean

Abstract

Abstract In this paper, we present the best possible parameters p and q such that the double inequality M p ( a , b ) 0 $a, b>0$ with a ≠ b $a\neq b$ , where M r ( a , b ) = [ ( a r + b r ) / 2 ] 1 / r $M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ ( r ≠ 0 $r\neq0$ ) and M 0 ( a , b ) = a b $M_{0}(a,b)= \sqrt {ab}$ is the rth power mean and V ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / 2 a b ) ] $V(a,b)=(a-b)/[\sqrt{2}\sinh^{-1}((a-b)/\sqrt{2ab})]$ is the second Yang mean.

Keywords

• power mean
• second Yang mean
• arithmetic mean
• quadratic mean
• geometric mean
• Lehmer mean