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Inequalities for means. (English) Zbl 0802.26009
The authors prove inequalities for power means, Stolarsky means, the geometric mean, and Gauss’s arithmetic-geometric mean and establish connections to elliptic integrals. A typical result is that \(((x^ t - y^ t)/(\ln (x/y)t))^{1/t}\) is a continuous function of \(t\) and strictly increases from \((xy)^{1/2}\) to \(\max (x,y)\) as \(t\) grows from 0 to \(\infty\).

MSC:
26D15 Inequalities for sums, series and integrals
33E05 Elliptic functions and integrals
26D07 Inequalities involving other types of functions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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