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An arithmetic-geometric mean inequality. (English) Zbl 1135.26302
Summary: Several integrals which are related to the arithmetic-geometric mean are developed and proved in a very elementary way. These results can be used to prove a known inequality which relates this mean to the logarithmic mean.

26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
[1] Gauss, C. F.: Werke. 3 (1866--1933)
[2] Salamin, E.: Computation of ${\pi}$ using the arithmetic-geometric mean. Math. comp. 30, 565-570 (1976) · Zbl 0345.10003
[3] Carlson, B. C.; Vuorinen, M.: SIAM review, problem 91--17. 34, 653 (1992)
[4] Vananamurthy, M. K.; Vuorinen, M.: Inequalities of means. J. of mathematical analysis and applications 183, 155-166 (1994) · Zbl 0802.26009
[5] Borwein, J. M.; Borwein, P. B.: PI and the AGM- A study in analytic number theory and computational complexity. (1987) · Zbl 0611.10001
[6] Almkvist, G.; Berndt, B.: Gauss, landen, Ramanujan, the arithmetic-geometric mean, ellipses,$ {\pi}$, and the ladies diary. Amer. math. Monthly 95, 585-607 (1988) · Zbl 0665.26007