Properties Of A Class Of Close-to-Convex Functions And The First Order Differential Subordination Of Analytic Functions

Analytic functions are important objects in the study of complex analysis,and they have important application in engineering,physics,electron,medicine,pneumatics and many branch-es of mathematics.In fact,the theory of analytic functions has gone deep into many branches of mathematics,such as differential equation,integral equation,probability theory and number theory,and it also promotes their development.This paper studies the properties of a class of close-to-convex functions and the first order differential subordination of analytic functions.The main contents are arranged as follows:In the second chapter,we discuss the class R which consist of analytic functions satisfying the condition Re e^i?₀f??z?>0 for some ?₀?[0,2?] in |z|<1,and it is a generalization of the classof analytic functions whose derivative has a positive real part.We study the following properties of functions in R:distortion theorem,the radius of convexity,close-to-convexity of the partial sums,extremal functions,functions with initial zero coefficients,radius of convexity of partial sums.We also study the radius of convexity of partial sums for functions in the subclass of the classdenoted bysatisfying the condition |f??z?-1|<1 in |z|<1.The classis a subclass of R,while Ris also a class of close-to-convex functions and it has many good properties.This research has the important theoretical significance.In the third chapter,we study the first order differential subordination of analytic functions.Suppose that p?z? is analytic function and satisfy p?0?=1 in |z|<1,we give a bound on ? so that implies that the function p?z? is subordinate to several well-known starlike functions.We also investigate that the subordination of p?z?when 1+?zp'?z?/p?z?,1+?zp'?z?/p²?z? are subordinate to several well-known starlike functions,respectively.By giving a new class of analytic functions,we establish the new differential subordinations.Our obtained result improves and generalizes the previous results.