# Inequalities for Means in Two Variables

@article{Alzer2003InequalitiesFM,
title={Inequalities for Means in Two Variables},
author={H. Alzer and S. Qiu},
journal={Archiv der Mathematik},
year={2003},
volume={80},
pages={201-215}
}
• Published 2003
• Mathematics
• Archiv der Mathematik
• 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… CONTINUE READING
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