In this thesis, we study the analytic properties of some special functions in quasiconformaltheory , such as Gaussian hypergeometric functions, complete elliptic integrals, generalizedelliptic integrals, distortion functions ?K(r),λ(K),ηK(t) and their generalizations, and es-tablish some sharp inequalities. By these results, the explicit quasiconformal Schwarz lemmaand some known estimates for the solutions to Ramanujan modular equations are improved.Moreover, comparison theorems between Sei?ert mean and Lehmer mean, generalized Heronmean and one-parameter mean are discussed.This thesis is divided into four chapters.In Chapter 1, we introduce the research background of this thesis and some concepts,notation and some known results used afterwards.In Chapter 2, a conjecture concerning Gaussian hypergeometric functions is solved by asimple method. Then, we prove some monotonicity theorems in terms of the combinationsof E(r) and elementary functions and obtain some sharp bounds for E(r). An asymptoticproperty of Ka(r) when râ†'1 is presented.In Chapter 3, at the beginning of the chapter, we prove some monotonicity theoremsof certain combinations in terms of the Hu¨bner function and elementary functions, presentsome new bounds for the Hu¨bner function, and obtain improvements upon known bounds forthe Hersch-P?uger distortion function, thus improving the explicit quasiconformal Schwarzlemma and some known estimates for the solutions to Ramanujan modular equations. Next,we obtain the bounds of the generalized Hu¨bner funciton ma(r) + log r which are given bythe complete elliptic integrals of the second kind E(r). By this connection, several sharpinequalities are obtained for the generalized Hersch-P?uger distortion function ?aK(r). Somenew properties and inequalities ofλ(K) andηK(t) are revealed, too.In Chapter 4, the best possible Lehmer mean bounds for Sei?ert means and the bestpossible one-parameter mean bounds for generalized Heron means are obtained. |