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Two inequalities for means. (English) Zbl 0615.26015
Let $$x>0$$, $$y>0$$, $$x\neq y$$, $$G(x,y)=(xy)^{1/2}$$, $L(x,y)=(x-y)/(\ln x-\ln y),\quad I(x,y)=\exp [-1+(x \ln x-y \ln y)/(x-y)].$ Continuing his work in Anz. Österr. Akad. Wiss., Math.-Naturwiss. Kl. 1986, 5-9 (1986; Zbl 0601.26014)], the author offers a proof of $[G(x,y)I(x,y)]^{1/2} < L(x,y) < (G(x,y)+I(x,y)).$
Reviewer: J.Aczél

##### MSC:
 26D15 Inequalities for sums, series and integrals 26A48 Monotonic functions, generalizations
Zbl 0601.26014