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 π. The authors study the monotoneity and
convexity properties of these quantities and obtain sharp inequalities for
them.