Inequalities for Means in Two Variables

Abstract.

We present various new inequalities involving the logarithmic mean $L(x,y)=(x-y)/(\log{x}-\log{y})$, the identric mean $I(x,y)=(1/e)(x^x/y^y)^{1/(x-y)}$, and the classical arithmetic and geometric means, $A(x,y)=(x+y)/2$ and $G(x,y)=\sqrt{xy}$. In particular, we prove the following conjecture, which was published in 1986 in this journal. If $M_r(x,y)= (x^r/2+y^r/2)^{1/r}(r\neq{0})$ denotes the power mean of order r, then $M_c(x,y)(\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})}$ with the best possible parameter $c=(\log{2})/(1+\log{2})$.