Some Properties Of Two Kinds Of Genceralized Elliptic Integrals

The Guassian hypergeometric function F(a,b;c;x)plays an important role in geometric function theory,quasiconformal theory,number theory and some other fields of mathematics,physics and other disciplines,as well as in engineering.And many other special functions in mathematical physics are particular cases or limiting of F(a,6;c;x).It is quite significant for us to study the properties of the hypergeometric function F(a,b;c;x)and its special cases.For example,these functions are essential in the studies of quasiconformal theory and the generalized Ramanujan's modular equations.In this thesis,we reveal some analytic properties of the Guassian hypergeometric func-tions,the generalized elliptic integral ?a,(r)and ?a'(r),and the generalized(p,q)-elliptic integrals ?p,q(r)and ?p,q(r),from which some new sharp inequalities follow.This thesis is divided into three chapters.In Chapter 1,we introduce the background of this thesis,some needed concepts,nota-tions and some known results.In Chapter 2,we show some monotonicity and concavity properties of certain combi-nations defined in terms of the zero-balanced hypergeometric function F(a,b;a + b;x)and elementary functions,thus extending some known properties of ?(r)and ?(r)to the zero-balanced hypergeometric functions and obtaining some properties of ?a(r),including sharp inequalities.In Chapter 3,we extend some known properties of ?(r)and ?(r)to ?p,q(r)and ?p,q(r),by studying the analytic properties of certain combinations defined in terms of ?p,q(r),?p,q(r)and some elementary functions.