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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.

##### MSC:
 33E05 Elliptic functions and integrals 33C05 Classical hypergeometric functions, $${}_2F_1$$ 26D15 Inequalities for sums, series and integrals 11F03 Modular and automorphic functions
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