Inequalities for means. (English) Zbl 0802.26009

Vamanamurthy, M. K.; Vuorinen, M.

J. Math. Anal. Appl. 183, No. 1, 155-166 (1994).

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 \(\infty\).

Reviewer: J.Aczél (Waterloo / Ontario)

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

Keywords:

Bernoulli-de l’Hospital rule; inequalities; power means; Stolarsky means; geometric mean; Gauss’s arithmetic-geometric mean; elliptic integrals