On The Best Possible Case Of Kind Of Upper Bounds For The φ_κ-Distortion Function

It is well known that special functions,such as Gaussian hypergeometric functions,elliptic integrals and distortion functions,have important applications in many fields of mathemat-ics,as well as in some other subjects and engineering.Elliptic integrals of the first and second kind κ(r) and ε(r),and the other special functions,such as M(r), Hersch-Pfluger distortion function, are some of the most important quasiconformal special functions. Quasiconformal Schwarz lemma is one of the most important properties of quasiconformal mappings,which says that Hersch-Pfluger distortion function φκ(r) gives the sharp bounds of a K-quasiconformal mapping of the unit disk B2into itself with∫(0)=0.Many distortion properties of quasicon-formal mappings are expressed in terms of φκ(r). And it also has applications in some other mathematical fields.Particularly,it appears in theory of Ramanujan’s modular equations and singular values of elliptic integrals.On the other hand,some properties of φκ(r),especially the sharp bounds for φκ(r),depend on those of the Hiibner function M(r).In order to obtain sharp bounds for the function φκ(r),one need to derive the better estimates of M(r) in terms of elementary functions.Hence,it is significant to obtain new properties for M(r),including sharper lower and upper bounds.In this thesis,the author studies two approximations for complete elliptic integrals by showing monotoneity and concavity of certain combinations of complete elliptic integrals and some elementary functions,and derives sharp functional inequalities for these special functions.On the other hand,the author studies a problem on the best possible case of a kind of lower and upper bounds in terms of simple elementary functions for the Hubner upper bound function M(r),obtain the lower and upper bounds for the constant max{c M(r)<(r’)c log4for all r∈(0,1)},and prove that min{c:M{r)>(1-r)clog4for all r∈(0,1)}=1.Applying these results,the author improves the known estimates for a kind of upper bounds of the Hersch-Pfluger distortion function φκ(r) in quasiconformal theory,thus improving a kind of the explicit quasiconformal Schwarz lemma.This thesis can be divided into four chapters.In chapter1, we introduce the concepts and notation of Gaussian hypergeometric func-tion F(a, b; c; x),complete elliptic integrals and their generalizations, the Hiibner function,Hersch-Pfluger distortion function and their developing history.In chapter2,we study two approximations for complete elliptic integrals by showing monotoneity and concavity of certain combinations of complete elliptic integrals and some elementary functions,and derive sharp functional inequalities for these special functionsIn chapter3,we study a problem on the best possible case of a kind of lower and upper bounds in terms of simple elementary functions for the Hubner upper bound function M(r),and improv the known estimates for a kind of upper bounds of the Hersch-Pfluger distortion function φκ(r) in quasiconformal theory,thus improving a kind of the explicit quasiconformal Schwarz lemma.