葉枝宏,何萍

(1.思茅師范高等專(zhuān)科學(xué)校,云南 普洱 665000;2.紅河學(xué)院數(shù)學(xué)學(xué)院,云南 蒙自 661100)

亞純函數(shù)的一類(lèi)值分布問(wèn)題

葉枝宏1,何萍2

(1.思茅師范高等專(zhuān)科學(xué)校,云南 普洱 665000;2.紅河學(xué)院數(shù)學(xué)學(xué)院,云南 蒙自 661100)

本文利用筆者在文獻(xiàn)[3]中構(gòu)造的亞純函數(shù)否定了一類(lèi)特殊的亞純函數(shù)退化為有理函數(shù),同時(shí)用Albert Edrei在文獻(xiàn)[4]中所構(gòu)造的亞純函數(shù)的性質(zhì)否定了一個(gè)整函數(shù)的值分布問(wèn)題推廣到亞純函數(shù)的可能.

有理函數(shù);亞純函數(shù);角域;射線;級(jí)與下級(jí)

0 引言

對(duì)于判定整函數(shù)f∞(z)何時(shí)退化為多項(xiàng)式方面,文獻(xiàn)[1]中介紹:如果存在一無(wú)界數(shù)列(z)=ωn}位于一條從原點(diǎn)出發(fā)的射線上,此時(shí)f(z)為次數(shù)不超過(guò)2的多項(xiàng)式.在1993年,喬建永在文獻(xiàn)[2]中得到:若對(duì)某自然數(shù)m有且存在一無(wú)界數(shù)列使得位于上,這時(shí)f(z)是次數(shù)不超過(guò)2m的多項(xiàng)式.

在考察亞純函數(shù)情形時(shí),自然會(huì)提出下述問(wèn)題:

1 主要結(jié)論

為回答這一問(wèn)題,本文得到如下定理:

定理1若f(z)為下級(jí)有窮的超越亞純函數(shù),存在一無(wú)界數(shù)列使得f(z)=ωn的根只位于Ω(-ε0,ε0)內(nèi).(這里ωn依賴(lài)于

證明:考察下級(jí)為零級(jí)的亞純函數(shù)(參見(jiàn)筆者在文獻(xiàn)[3]中的詳細(xì)介紹)

對(duì)于該函數(shù)而言,在Ω(-ε0,ε0)外,若取定充分小ε0(0＜ε0＜1),同時(shí)考慮角域C/Ω上函數(shù)f(z)的模.其Ω=Ω(-ε0,ε0).

此時(shí)對(duì)于?z?C/Ω和一切的k?N有

p=b1sinε0≤|z-bk|

2 結(jié)論的應(yīng)用

對(duì)于問(wèn)題,Albert Edrei在文獻(xiàn)[4]中介紹了函數(shù)

該函數(shù)顯然是極點(diǎn)只位于正實(shí)軸上的亞純函數(shù).若取定一無(wú)界正實(shí)數(shù)列,此時(shí)對(duì)于Im z≠0,有

[1]Baker,In,&Liverpool l.s.o.The value distribution of entire funtions of order atmost one[J].Acta Sci,Math.,41(1979):3-14.

[2]喬建永.整函數(shù)的值分布與集[J].數(shù)學(xué)學(xué)報(bào),1993,36(3):418-422.

[3]葉枝宏.零點(diǎn)和極點(diǎn)位于有限條射線上的亞純函數(shù)[J].云南師范大學(xué)學(xué)報(bào),2007,1:101-103.

[4]Albert Edrei.Meromorphic functions with three radially distributed values[J].Amerrica Mathematical Society,1955,78(2):277-293.

The Problem s of The Value Distribution about Meromorphic Functions

YE ZhiHong1,HE Ping2
(1.Teacher College of Mathematics,Simao 665000,China;2.Department of Mathematics,Honghe University,Mengzi 661100,China)

In this paper,we use a function which is structured in[3]and deny meromorphic Functions to degenerate rational function.At the same time,the value distribution of entire functions to extend meromorphic functions is denied by a function which was constructed by Albert Edrei in[4]

Rational Functions;Meromorphic Functions;Angular Domain;Rays;Order and Lower Order

0174.52

A

1008-9128(2010)02-0034-02

2009-11-30

紅河學(xué)院博士、碩士科研啟動(dòng)項(xiàng)目(XSS08013)

葉枝宏(1980-),男,云南普洱人,碩士,助教.研究方向:復(fù)分析.

[責(zé)任編輯 宋煥斌]