Two inequalities for means. (English) Zbl 0615.26015

Alzer, Horst

C. R. Math. Acad. Sci., Soc. R. Can. 9, 11-16 (1987).

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

Keywords:

generalized logarithmic mean; inequalities between means; geometric mean; identric mean; arithmetic mean; increasing differentiable functions

Citations:

Zbl 0601.26014