周 茜

(江蘇食品職業(yè)技術(shù)學(xué)院 基礎(chǔ)教學(xué)部, 江蘇 淮安 223003)

某些亞純P-葉函數(shù)的性質(zhì)

周 茜

(江蘇食品職業(yè)技術(shù)學(xué)院 基礎(chǔ)教學(xué)部, 江蘇 淮安 223003)

在定義新的線性算子Lp(a,c) 的基礎(chǔ)上,研究得到某些亞純P-葉函數(shù)的性質(zhì)．所得主要結(jié)論為一個(gè)定理及兩個(gè)推論．

解析函數(shù); 亞純P-葉函數(shù); 線性算子;

0 引言

本文首先對(duì)U0={z:z∈c,0<|z|<1}=U{0}內(nèi)解析的P-葉函數(shù)的全體組成的函數(shù)類、線性算子Lp(a,c)、一個(gè)函數(shù)類做出定義,并在引理1的理論基礎(chǔ)上研究討論某些亞純P-葉函數(shù)的性質(zhì)．

(1)

且在U0={z:z∈c,0<|z|<1}=U{0}內(nèi)解析的P-葉函數(shù)的全體組成的函數(shù)類．

Lp(a,c)f(z)=φp(a,c;z)*f(z)

(2)

可證得[2]

(3)

z(Lp(a,c)f(z))′=aLp(a+1,c)f(z)-(a+p)Lp(a,c)f(z)

(4)

由線性算子Lp(a,c)定義一個(gè)函數(shù)類:

(5)

(6)

為了建立我們的主要結(jié)論,需以下引理．

1 主要結(jié)論

(7)

(8)

則q(z)在U內(nèi)解析,且q(0)=1．

若設(shè)

(9)

則Re{h(z)}>δ．

由式(4),得

(10)

設(shè)函數(shù)

(11)

{Ψ(q(z),zq′(z));z∈U}?Ω={w∈C:Re(w)>α},

因此,對(duì)一切z∈U,Ψ(ir2,s1)?Ω,由引理可得

定理得證．

(12)

則

(13)

證明因

又因λ≥1,由式(12)及式(7)(μ=1),可得

Re{(1-λ)(zpLp(a,c)f(z))μ+λzpLp(a+1,c)f(z)·(zpLp(a,c)f(z))μ-1} >α,

(0≤α<1;μ>0;p∈N;z∈U)

則

(14)

Re{(1-λ)zpLp(a,c)f(z)+λzpLp(a+1,c)f(z)}>α(z∈U),

則

(15)

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[3] Liu J L, Srivastava H M. A linear operator and associated families of meromorphically multivalent functions?[J].J Math Anal Appl, 2001, 259(2):566-581.

[4] Miller S S, Mocanu P T. Second order defferential inequalities in the complex plane?[J]. J Math Anal Appl, 1978, 65(2):289-305.

[5] Miller S S, Mocanu P T. Differential subordinations and univalent functions?[J].Michigan Math J,1981,28(2):157-171.

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SomePropertiesofMeromorphicP-valentFunctions

ZHOU Qian

(Jiangsu Food Science College, Huaian Jiangsu 223003, China)

Definition is presented in this paper, on the basis of the new linear operator, get some research and the properties of the pureP-leaf function. The main conclusion is a theorem and two inferences

analytic functions; meromorphicp-valent functions; linear operator

2013-07-16

周茜(1981-), 女, 江蘇沭陽人, 講師, 碩士， 研究方向?yàn)閺?fù)分析.

O174.3

A

1671-6876(2013)04-0293-03

[責(zé)任編輯李春紅]