Some Properties Of Generalized Distortion Functions And Means

In this thesis, we shall extend some properties and inequalities for special functions in quasiconformal theory, such as Agard distortion function ?x(t) and linear distortion function ?(K), to the generalizations of these function. In addition, the Schur quadratic concavity of the first Neuman mean, and comparison theorems for the second Neuman mean and logarithmic mean, two Seiffert means, Neuman-Sandor mean are proved.This thesis can be divided into three chapters.In Chapter 1, we introduce the research background of this thesis, several concepts, notation and some known results used afterwards.In Chapter 2, an exponential inequality for linear distortion function ?(K) is established, and some monotonicity properties of certain combinations in terms of generalized Agard distortion function ?K(a,t) and elementary functions are presented, thus generalizing several well-known inequalities for ?K(t) and ?(K).In Chapter 3, some necessary and sufficient conditions for the Schur quadratic concavity of the first Neuman mean are obtained, and several sharp inequalities are presented for the second Neuman mean.