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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
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