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Generalized elliptic integrals and modular equations. (English) Zbl 0951.33012
Authors’ summary: In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to $\pi$. The authors study the monotonicity and convexity properties of these quantities and obtain sharp inequalities for them.
Reviewer: G.D.Anderson (East Lansing)

33E05Elliptic functions and integrals
33C05Classical hypergeometric functions, ${}_2F_1$
26D15Inequalities for sums, series and integrals of real functions
11F03Modular and automorphic functions
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