与函数类有关的积分表示和卷积性质

彭 娟,刘文娟,杨 清

(扬州大学 数学科学学院,江苏 扬州 225002)

彭 娟,刘文娟,杨 清

(扬州大学 数学科学学院,江苏 扬州 225002)

亚纯函数; 微分从属; 卷积; 积分表示

0 引言

(1)

且在去心单位圆U*={z:z∈C,0lt;|z|lt;1}=U{0}内p叶解析的函数f(z)组成的函数类.

我们定义如下函数φp(a,c;z):

(2)

(3)

(4)

从(1)和(4)可看出,

(5)

由(5)有

(6)

(7)

(8)

定义1 设f(z)∈Σp,若满足从属条件

特别地∂=0,有

为了证明主要的结论,我们需要以下的引理.

引理1[10]设β,γ∈C,h(z)在U内解析,凸单叶,并且

h(0)=1,R(βh(z)+γ)gt;0(z∈U).

如果p(z)在U内解析,p(0)=1,若满足

则

p(z)h(z)(z∈U).

(9)

进一步地,若

则

证明由

(10)

(11)

则

(12)

令

(13)

(14)

1 主要结果

(15)

其中ω(z)在U内解析,ω(0)=0,|ω(z)|lt;1(z∈U).

(16)

其中ω(z)在U内解析,ω(0)=0,|ω(z)|lt;1(z∈U).

(17)

(18)

对(18)两边积分有

(19)

则

(20)

其中ω(z)在U内解析,ω(0)=0,|ω(z)|lt;1(z∈U).

其中ω(z)在U内解析,ω(0)=0,|ω(z)|lt;1(z∈U).

(21)

其中ωj(z)(j=1,2)在U内解析,ωj(0)=0,|ωj(z)|lt;1(z∈U;j=1,2).

(22)

其中ω1(z)在U内解析,ω1(0)=0,|ω1(z)|lt;1(z∈U).

(23)

(24)

其中ωj(z)(j=1,2)在U内解析,ωj(0)=0,|ωj(z)|lt;1(z∈U;j=1,2).

对(24)两边积分有

其中ω(z)在U内解析,ω(0)=0,|ω(z)|lt;1(z∈U).

致谢作者感谢刘金林老师的悉心指导!

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[责任编辑：李春红]

PENG Juan,LIU Wen-juan,YANG qing

(School of Mathematical Science,Yangzhou University,Yangzhou Jiangsu 225002,China)

meromorphic functions； differential subordination； integral representation； integral operator

O174.5

A

1671-6876(2012)01-0022-04

2011-12-12

彭娟(1986-),女,湖北洪湖人， 硕士研究生,研究方向为复分析.