Generalized Elliptic Integrals And Ramanujan Mode Equations Solution Nature

It is well known that the Gaussian hypergeometric function F(a,b; c; x), com-plete elliptic integrals K(r) and E(r), generalized elliptic integrals Ka(r) and Ea(r),generalized Hersch-P?uger distortion function ?K(a,r) and other related specialfunctions play an extensive and important role in number theory, quasiconformaltheory, geometry and many other areas of mathematics and other disciplines.As one kind of the most important special functions, the generalized ellipticintegrals are important special cases of hypergeometric functions. On the otherhand, they are the generalization of the complete elliptic integrals. Moreover, thegeneralized Gr¨otzsch ring functionμa(r), generalized Hu¨bner upper bound functionma(r), generalized Hersch-P?uger distortion Function ?K(a,r), Agardη-distortionfunctionηK(a,t) and linear distortion functionλ(a,K), which appear in the gener-alized modular equation, are defined in terms of the generalized elliptic integrals. Infact, one can obtain the properties ofμa(r), ma(r), ?K(a,r),ηK(a,t) andλ(a,K)by studying the properties of the generalized elliptic integrals. In particular, theestimates of the function ?K(a,r) given by elementary functions often depend onthe analytic properties of certain combinations of the functionsμa(r), ma(r) andsome elementary functions. Thus, the researches on the properties and applicationsof the generalized elliptic integrals are significant.In this thesis, we extend some important properties of the complete ellipticintegrals K(r) and E(r) to the functions Ka(r) and Ea(r), reveal some new analyticproperties of Ka(r) and Ea(r) by studying the properties of the combination of thegeneralized elliptic integrals Ka(r), Ea(r) and some elementary functions, from whichsome functional inequalities and better estimate for Ka(r) and Ea(r) follow. weshall also derive some inequalities of the functionsμa(r) and ma(r) by studying themonotonicity,convexity and concavity of certain combinations of the functionsμa(r),ma(r) and elementary functions, by which we strengthen the upper and lower boundsof the generalized Hersch-P?uger distortion ?K(a,r), quasiconformal Schwarz lemmaand the solution of generalized Ramanujan's modular equation.This paper is divided into four chapters as follows:In the first chapter,we introduce the research background of this thesis and some concepts, notation and some known results used afterwards.In the second chapter, we present some analytic properties of certain combina-tions of functions Ka(r), Ea(r) and elementary functions, and obtain some functionalinequalities. Then, we study the monotonicity, convexity and concavity of somefunctions defined in terms of the functions Ka(r) and Ea(r), get some functionalinequalities for them, and improve some known bounds of Ka(r) and Ea(r). Weshall also obtain some new analytic properties of the generalized elliptic integralsby studying the dependence on parameter a of the generalized elliptic integrals.In the third chapter, we study some analytic properties of the generalizedGro¨tzsch ring functionμa(r), the function ma(r) and the related generalized ellipticintegrals, and obtain some inequalities of the functionsμa(r) and ma(r). Then, weapply some results in the second chapter and the relation between ?K(a,r) andμa(r) and ma(r), to get some better estimates of the function ?K(a,r), which aregiven in terms of certain elementary functions.In the fourth chapter, we show some analytic properties of the solution ?K(a,r)to the generalized Ramanujan's modular equation, the generalized Agardη-distortionfunctionηK(a,t) and generalized linear distortion functionλ(a,K).