Some Properties Of Generalized Elliptic Integrals And Generalized ¦Ç-distortion Function

The generalized elliptic integrals have many applications in some mathematical fields,such as number theory, quasiconformal theory and geometry. It is well known that thegeneralized elliptic integralsΚ_a(r),ε_a(r), the related special functions m_a(r) and the so-lution of Ramanujan modular equationφΚ(a,r) reduce to the complete integrals of thefirst and the second kindΚ(r), E(r), m(r) and Hersch-Pfluger distortion functionφK(r),respectively, when a = 1/2. The properties of the solutionφΚ(a,r) of Ramanujan mod-ular equation are closely related to the properties ofηΚ(a,t) andλ(a,Κ), which playimportant roles in quasiconformal mappings, quasiregular mappings, quasisymmetricfunctions and in other mathematical fields. So it is significant to study the monotonic-ity, concavity and convexity properties ofΚ_a(r),ε_a(r) and the generalizedη-distortionfunction.This thesis is divided into three chapters as follows.In the first chapter, we introduce some definitions and notation used in this thesis.In the second chapter, we present some monotonicity and convexity properties ofcertain functions defined in terms of the generalized elliptic integralsΚ_a(r) andε_a(r),from which some inequalities follow.In the third chapter, we show some properties of the functionsηΚ(a,t) andλ(a,Κ)defined in terms of the solutionφΚ(a,r)of Ramanujan modular equation.