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Inequalities for means in two variables. (English) Zbl 1020.26011
Summary: 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 [H. Alzer, Arch. Math. 47, 422-426 (1986; Zbl 0585.26014)]. If $$M_r(x,y) = (x^r/2+y^r/2)^{1/r}$$ ($$r\neq 0$$) denotes the power mean of order $$r$$, then $M_c(x,y) <\frac 12(L(x,y)+I(x,y)) \qquad (x,y>0, x\neq y)$ with the best possible parameter $$c=(\log 2)/(1+\log 2)$$.

##### MSC:
 26D15 Inequalities for sums, series and integrals 26E60 Means
Zbl 0585.26014
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