Special Functions In Qusiconformal Mappings And Restrained Sequences Of M(?)bius Transformations In Higher Dimensions

As is well known, special functions in qusiconformal mappings, such as Gaus-sian hypergeometric function F(a,b; c; x), complete elliptic integrals of the first and second kinds K(r) and ε(r), Grotzsch ring function μ(r), Hersch-Pfluger distortion function φk(r), have important applications in quasiconformal mappings, number theory, the theory of means and other fields of the mathematics and mathematical physics.The Gaussian hypergeometric function is one of the most important classes of special functions, many other special functions and elementary functions are only its special cases or limiting cases. In 1980s, by use of the Gaussian hypergeometric function, De Branges completed the proof of the famous Bieberbach Conjecture. This shows that it is significant to study the properties of hypergeometric functions. The complete elliptic integrals, as the special cases of the Gaussian hypergeomet-ric function, are closely associated with some problems in the geometry function theory, such as the estimation of the perimeter of an ellipse, the computations of Grotzsch ring function μ(r) and circumference ratio Ï€, and conformal mappings of the polygons. Therefore, to establish some new properties and inequalities for the complete elliptic integrals is helpful to proceed the development of the geometry function theory. Moveover, by proving some sharp inequalities for Hersch-Pfluger distortion function φk(r), the explicit quasiconformal Schwarz lemma and the estimates for the solutions to Ramanujan modular equations can be improved.In this thesis, we study some analytic properties of these special functions in quasiconformal mappings and their generalizations, and establish some identi-ties and sharp inequalities. With their applications, some well-known results can be proved and extended. Moreover, the restrained sequences of Mobius transfor-mation in higher dimension, as the general convergence sequences of continued fractions, are discussed in hyperbolic geometry.This thesis can be divided into four chapters.In Chapter 1, we introduce the research background of this thesis and some concepts, notation and some known results used afterward.In Chapter 2, we firstly establish the necessary and sufficient condition for the convexity and concavity of the complete elliptic integrals of the first and second kinds with respected to Holder means. Next, an optimal Holder mean inequality for the complete elliptic integrals is proved. Besides, making use of Toader mean, we present several sharp inequalities for ε(r) or the perimeter of an ellipse, which improve some well-known bounds. Finally, by looking for the series expansion of Ramanujan constant function, an asymptotic bound of the generalized elliptic integrals of the first kind Ka(r)(râ†'1) is given.In Chapter 3, at the beginning of the chapter, we establish a criterion for the monotonicity of the quotient of power series, as applications, the Ramanujan cubic transformation inequalities for zero-balanced hypergeometric function are proved. Next, we study the quotient of hypergeometric functions μa*(r) (generalized Grotzsch function) in the theory of Ramanujan’s generalized modular equation, find an infinite product formula for μ1/3*(r) and obtain some inequalities for μa*(r) by Ramanujan cubic transformation inequalities. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan’s cubic transformation.In Chapter 4, with the Clifford algebra, we give some analytic and geometric characterizations of the restrained sequences of Mobius transformation in higher dimensions.