Optimal Inequalities among Various Means of Two Arguments

Abstract

We establish two optimal inequalities among power mean $M_p\left(a,b\right)={\left(a^p/2+b^p/2\right)}^{1/p}$, arithmetic mean $A\left(a,b\right)=\left(a+b\right)/2$, logarithmic mean $L\left(a,b\right)=\left(a-b\right)/\left(\log a-\log b\right)$, and geometric mean $G\left(a,b\right)=\sqrt{ab}$.