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Inequalities for certain means in two arguments. (English) Zbl 0938.26011

Let $G=\sqrt{ab}$; $L=\left(b-a\right)/\left(lnb-lna\right)$; $I=\frac{1}{e}{\left({b}^{b}/{a}^{a}\right)}^{1/\left(b-a\right)}$; $A=\frac{a+b}{2}$, $S={a}^{a/\left(a+b\right)}{b}^{b/\left(a+b\right)}$. The following inequalities are valid:

${A}^{2}/I<\left(4{A}^{2}-{G}^{2}\right)/3I
$AL+SI<2{A}^{2}<{S}^{2}+{G}^{2},$
$\left(4{A}^{2}-2{G}^{2}\right)/e
${\left(S/A\right)}^{2}<{\left(I/G\right)}^{3},$
$\left({A}^{2}-{G}^{2}\right)/{A}^{2}
$\left(S-G\right)/\left(S-A\right)>\sqrt{2}·$

MSC:
 26D15 Inequalities for sums, series and integrals of real functions 26E60 Means