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Monotonicity theorems and inequalities for the complete elliptic integrals. (English) Zbl 1059.33029
Summary: We prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind, $\Cal K$ and $\Cal E$. These results lead to sharp symmetrical bounds for $\Cal K$ and $\Cal E $, which improve recently discovered inequalities.

MSC:
33E05Elliptic functions and integrals
26D15Inequalities for sums, series and integrals of real functions
33C05Classical hypergeometric functions, ${}_2F_1$
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References:
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