一类P叶解析函数

A CLASS OF P-VALENT ANALYTIC FUNCTIONS

设Φ_p(n,A,B)表示在单位圆盘E内满足从属关系((D_p^nf(z))′)/(z^(p 1))<p((1+Az)/(1+Bz))的解析函数f(z)=z^p+sum from k-1 to (x)a_(p-k)z^(p-k)组成的类,这里-1≤B<A≤1,p是正整数,n是非负整数,D_p^0f(z)=f(z),D_p^(n+1)f(z)=(z^p)/((1-z)~2)*D_p^nf(z)。本文证明Φ_p(n,A,B)中的函数是P叶的。进而我们得到保类积分算子,精确系数估计和闭包定理。另外通过系数得到函数f(z)在类Φ_p(n,A,B)中的一个充分条件。

Let p(n,A,B) denote the class of functions f(z)=which areanalytic in the unit disc E and satisfy the conditionwhere p is a positive integer, n is any nonnegative integer,andWe show that the functions in are p-valent. Thenwe obtain class preserving integral operators, sharp coefficient estimates and a closure theorem for the class (n.A.B). We also obtain a sufficient condition, in terms of coefficients, for a function to be in (n.A,B) when B<0.