The Properties And Applications Of Complete(p,q)-Elliptic Integrals

The Gaussian hypergeometric function F(a,b;c;x),complete elliptic integrals K(r)and E(r),generalized elliptic integrals Ka(r)and Ea(r),complete(p,q)-elliptic integrals Kp,q(r)and Ep,q(r)and other related special functions have wide and important applications in geometric function theory,quasiconformal theory,Ramanujan's modular equations and many other fields of mathematics,and in physics and engineering as well.Complete(p,q)-elliptic integrals Kp,q(r)and Ep,q(r)are closely associated with(p,q)-Gr?tzsch ring function ?p,q(r),(p,q)-Hübner function Mp,q(r)and the related(p,q)-distortion functions ?Kp,q(r),?Kp,q(t)and?p,q(K).In the case when p=q=2,these functions reduce to K(r),E(r),?(r),M(r),?K(r),?K(t)and ?(K),respectively,which are called the special(quasiconformal)functions and not only are essential in the studies of distortion properties of quasiconformal mappings,but also have important applications in some other mathematical fields such as number theory.In this thesis,we mainly study and reveal some analytic properties of complete(p,q)elliptic integral Kp,q(r),(p,q)-Gr?tzsch ring function ?p,q(r)and(p,q)-Hübner function Mp,q(r).Then we shall apply these results to study a class of Ramanujan modular equations defined in term of complete(p,q)-elliptic integrals of the first kind,and present some analytic properties of the(p,q)-generalized distortion functions ?Kp,q(r),?Kp,q(t)and ?p,q(K).This thesis is divided into three chapters.In Chapter 1,we introduce the related concepts,notations and several known results,state the developing history of the complete(p,q)-elliptic integrals,and give the background of this thesis.In Chapter 2,we show the analytic properties of certain combinations defined in terms of Kp,q(r),Ep,q(r)and elementary functions,and extend some well-known results for the Grotzsch ring function ?(r)and Hubner function M(r)to ?p,q(r)and Mp,q(r)by revealing the monotonicity and convexity properties of certain combinations defined in terms of ?p,q(r),Mp,q(r)and elementary functions.In addition,several properties of the generalized elliptic integral Ka(r)of the first kind are obtained,too.In chapter 3,we obtain several properties of the solution ?Kp,q(r)of a class of Ramanujan modular equations defined by Kp,q(r),and the(p,q)-generalized distortion functions ?Kp,q(t)and?p,q(K).from which some sharp inequalities satisfied by ?Kp,q(r),?Kp,q(t)and ?p,q(K)follow.