Generalized elliptic integrals and modular equations

In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane o nto a parallelogram and are naturally related to Gaussian hypergeometric fu nctions. Certain combinations of these integrals also occur in analytic num ber theory in the study of Ramanujan's modular equations and approximations to pi. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them.