Generalized elliptic integrals and modular equations

Abstract

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 . The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them. 1 12 {() 1 6 + {(1 -)(1 -)} 1 6 } = 1.