一类近于凸函数子族的研究

A subclass of close-to-convex functions

函数g(z)<G(z),当且仅当存在单位开圆盘E内的解析函数w(z)∈B0,即满足:w(0)=0,|w(z)|<1,使得g(z)=G(w(z))(z∈E),设P[A,B]={P(z)∶P(0)=1,P(z)在E内解析且满足P(z)<11++ABzz,-1≤B<A≤1},一个函数g(z)∈C[A,B]当且仅当(zgg′′((zz)))′<11++ABzz.函数族Kβ′[A,B]={f(z)∶f(0)=f′(0)-1=0,f(z)在E内解析,g(z)∈C[A,B],且Re{zgf′(z(z))}>β,-1≤B<A≤1},这是近于凸函数的一个子集,从而这些函数是单叶的.利用Janowski介绍的函数类P[A,B]的性质,参考Khalida InayatNoor研究Cβ*[A,B]的方法,研究这个函数族系数估计和半径问题,同时讨论Kβ′[A,B]与其他单叶函数子族的关系.

A function g（z）〈G（z） if and only if it exited is a w（z）∈B0, where w（0）=0,｜w（z）｜〈1 , so that g（z）=G（w（z））（z∈E）.Let P[A,B], -1≤B〈A≤1, denoted the class of functions p（z） ： p（0）=1, analytic in the unit open disk E, such that p（z） was subordinate to 1＋Az/1＋Bz.A function g（z）was said to belong to the class C[A,B]if and only if （zg＇（z））＇/g＇（z） was in P[A, B], KB[A, B]-the class of analytic functions f（z）,where f（0）=f＇（0）-1=0, regular in E satisfying the condition. Re {zf＇（z）/g（z）}〉B, g∈C[A,B], -1≤B〈A≤1.These functions in this class were close-to-convex and hence univalent. With the help of the new Class P[-A,B]which introduced by Janowski, we studied its relationship with other functions. Co-efficient problem, distortion theorems, radius of convexity and other properties were solved.