Fully Elliptic Integrals And Related Special Function Nature

It is well known that special functions,such as Gaussian hypergeometric function, com-plete elliptic integrals,and generalized Gr(o|¨)tzsch ring function μa(r) have extensive and im-portant applications in number theory,quasiconformal theory,geometry and many other areasof mathematics,as well as in some other subjects and engineering. In this thesis, we studythe analytic properties of the special functions F (a, b; c, r),μa(r), K(a, r) and ηK(a, t), andestablish some sharp inequalities for them. As their applications, some well-known bounds ofrelated special function are improved.Besides,the explicit quasiconformal Schwarz lemma andthe estimates for the solutions to generalized Ramanujan modular equations are strength-ened.This thesis is divided into four chapters.In the first chapter,we introduce the research background of this thesis and some con-cepts,notation and known results needed afterwards.In the second chapter,we study some monotonicity and concavity properties of thecombinations defined in terms of gamma function Γ(x), its logarithmic derivative ψ(x),from which sharp inequalities for some important constants in quasiconformal theory areobtained.Moreover,a conjecture concerning geometric concavity of gamma function is provedto be untrue, and the complete answer is given.In the third chapter,we present some analytic properties of generalized elliptic integralsKa(r) and Ea(r). We also prove some monotonicity properties of certain functions definedin term of generalized Gr(o|¨)tzsch ring function μa(r) and some elementary functions,thusimproving the well known bounds of μa(r) given in terms of infinite product expansion aswell as inequalities for the relationship between μ(r) and μa(r).Furthermore,the lower boundof generalized Hersch-Pfluger distortion function is obtained.In the fourth chapter, we show some analytic properties of the solution K(a, r) tothe generalized Ramanujan's modular equation,and of the generalized Agard η distortionfunction ηK(a, t), from which several sharp inequalities are obtained.