Some Properties Of Generalized ?_?-distortion Functions,Generalized Hyperbolic Functions And Means

In this thesis, we study the analytic properties of some special functions in quasiconformal theory, such as Gaussian hypergeometric functions, complete elliptic integrals, distortion functions, and their generalizations, and obtain the monotonicity properties of the some new combinations in terms of the complete elliptic integrals and distortion functions. Further-more, we present several analytic properties of the generalized trigonometric and hyperbolic functions. Moreover, we study some mean functions, and obtain new inequalities.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, we show some basic properties of the generalized elliptic integrals ?_a(r) and ?_a(r) and the generalized distortion functions ?_?(a,r), and the monotonicity properties of the some new combinations in terms of ?_?(a, r) and elementary functions.In Chapter 3, some properties of the generalized trigonometric and hyperbolic functions are presented.In Chapter 4, we study the mean functions, compare some of them and get some inequal-ities.