The Special Functions In The Theory Of Quasiconformal Mappings

This thesis covers the topic of special functions related to distortion theory of quasiconformal mappings. These special functions include hypergeometric func-tions, elliptic integrals, distortion functions and elliptic functions, as well as their generalizations.In the first chapter, we first survey the history and development of the theory of special functions. Then we give a summary outlining the main results of the thesis. The innovations of this thesis are also illustrated. Finally, some notions and two simple and useful lemmas which are used in the thorough text are introduced in this chapter too.In the chapter2, Gaussian hypergeometric functions are studied. Hypergeo-metric functions are basic special functions, and many special functions may be written in terms of the hypergeometric function. We summarize some general properties of hypergeometric functions, including integral representation, behavior of hypergeometric functions at logarithmic singularities, contiguous relations and Elliott identity, which play an important role in the study of elliptic integrals. Next, we recall the monotonicity of some hypergeometric functions from which we obtain the behavior of singularities and linearizations of hypergeometric function-s. We also investigate the generalized convexity of hypergeometric functions and deduce multiplication property for hypergeometric functions.Elliptic integrals are special cases of hypergeometric functions. In the chap-ter3, we mainly study the generalized elliptic integrals with one parameter and a special function ma(r) which is defined as a product of two generalized elliptic integrals. The Legendre identity and Landen transformation formulae are recalled. The monotonicity property and the inequalities involving power means of the spe-cial function ma(r) with a=1/2are studied. Some of these properties are also generalized to the functions with one parameter. The monotonicity of the func-tion ma(r) plays an important role in the sharp estimates of the distortion function φK(r) and the modular functionφKa(r).The Grotzsch ring function which is the quotient of elliptic integrals represents the modulus of the plane Grotzsch ring. The Grotzsch ring function also appears in the classic modular equations. In the chapter4, we study the Grotzsch ring function and its generalization. By using the Landen transformation formulae we get similar transformation formulae for the generalized Grotzsch function. We also recall an infinite product for the Grotzsch ring function. After listing some well-known monotonicity properties for the Grotzsch ring function, we study the power mean inequality for it which is a generalization of the geometric mean inequality for the Grotzsch ring function. We prove some monotonicity properties for the generalized Grotzsch function which can be used to obtain some sharp estimates for the modular function and the generalized η—function. A power mean inequality involving two parameters for the generalized Grotzsch function is also obtained. We introduce a notion of exponential quasiadditivity, and as an application, we deduce a functional inequality for the generalized Grotzsch function.In the chapter5, we study the quasiconformal distortion functions and their generalizations. These functions have close connection with Ramanujan’s mod-ular equations. There are many algebraic identities which these transcendental functions satisfy. We introduce many important monotonicity properties and lin-earization of distortion functions. Then we prove the generalized convexity for the Hersch-Pfluger distortion function involving Holder means. Some sharp in-equalities for the modular function are established. These inequalities involve the generalized Grotzsch function and the special function ma(r). Combing the in-equalities obtained in the chapters3and4, we can get many elementary sharp bounds for the Hersch-Pfluger distortion function and the modular function. We investigate the connections between the generalized η-function and the generalized Grotzsch function and deduce some asymptotic sharp bounds for the generalized η-function by making use of the inequalities for the generalized Grotzsch function which come from chapter4. The generalized convexity and functional inequalities for the generalized η-function are studied too.Jacobian elliptic functions are the main topic of the chapter6. We recall the definitions and basic properties of Jacobian elliptic functions. We prove some monotonicity properties for elliptic functions from which some sharp inequalities are obtained and generalize the corresponding inequalities for the trigonometric functions and elliptic integrals. We prove the Holder concavity of the Jacobian elliptic sine function.In the last chapter, we make more generalizations of the functions we studied in the previous chapters by introducing more parameters. We make some basic investigations and obtain monotonicity properties for these generalized functions.