一类近于凸函数

A Class of Close-to-convex Functions

设函数f(z)在单位圆E={|z|＜i}上是解析的,且(f(z) f(z)/z)≠0(z∈E)．如果对某个a>0及0<β<1,存在E上的星像函数g(z), go=g'(o)-1,使得Rc[(1-a)z f'z/gz+(zf'z'/g'z]>β(z∈E)成立,则称f为β阶a-近似于凸函数．本文证明了β阶a一近似于凸函数是β阶近于凸的,还得到了β阶一近似于凸函数的分解定理及几个系数不等式.

Let function f(z) be analytic in the unit disk E={z:|z|<1}and (f(z) f(z)/z) ≠0 in E. We call f an a-close-to conver ordex β if for some α>0,0<β<1 there exists a starlike function g(z) in E with g(0) =g'(0) -1=0 such that Re ((l-α)+α >β (z∈E) g (z) +> β (z∈E) We have proved that all functions of a-close-to-convex of order β are close-to-convex of order β and have obtained a decomposition theorem as well as several coefficient inequalities for these functions.