李華仙

(暨南大學(xué)數(shù)學(xué)系,廣東廣州 510632)

代數(shù)體函數(shù)微分多項(xiàng)式的值分布

李華仙

(暨南大學(xué)數(shù)學(xué)系,廣東廣州 510632)

利用亞純函數(shù)的Nevanlinna值分布理論和方法,研究了一類代數(shù)體函數(shù)微分多項(xiàng)式的值分布問(wèn)題,推廣和改進(jìn)了相關(guān)文獻(xiàn)的結(jié)論.

代數(shù)體函數(shù);值分布;微分多項(xiàng)式;小函數(shù)

1 引言及主要定理

全文采用Nevanlinna值分布理論的通常記號(hào)[1].

若φ(x)是w的小函數(shù),即φ(x)是滿足T(r,φ)=S(r,w)的亞純函數(shù).設(shè)w(z)是v值代數(shù)體函數(shù).F(w)是關(guān)于w的代數(shù)體函數(shù),a∈C或a=∞,z0∈C,用τ(F(w),a,z0)表示F(w)在z0點(diǎn)取a值的重?cái)?shù),則定義

其中iα(α=0,1,···,n),s,t為非負(fù)整數(shù),ai(z),bj(z)(i=1,2,···,m,j=0,1,···,s-1),φ(z)是w的小函數(shù).

2 一些引理

3 定理1.2的證明

4 定理1.3的證明

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[3]龐學(xué)誠(chéng).代數(shù)體函數(shù)及其導(dǎo)數(shù)的Picard值[J].數(shù)學(xué)年刊,1993,1(1):51-56.

[4]Katajamaki K.Value distribution of some differential polynomials of entire algebroid functions[J].Complex Variables,1996,7:422-429.

[5]Gao L Y.On value distribution of differential monomial of algebroid functions[J].Indian J.Pure Appl. Math.,2003,34(5):799-804.

[6]黃斌.代數(shù)體函數(shù)及其微分多項(xiàng)式[J].數(shù)學(xué)學(xué)報(bào),2004,47(1):157-172.

[7]何育贊,肖修治.高階代數(shù)微分方程的單值亞純函數(shù)解和有限多分支解[J].中國(guó)科學(xué),1983,6:514-522.

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On value distribution of differential polynomial of algebroid functions

Li Huaxian

(Department of Mathematics,Jinan University,Guangzhou510632,China)

Using Nevanlinna theory of value distribution of algebroid functions,the value distribution problem of algebroid differential polynomial is studied,we improve and generalize the related results.

algebroid functions,the value distribution,differential polynomial,small function

O174.52

A

1008-5513(2013)03-0318-07

10.3969/j.issn.1008-5513.2013.03.014

2012-10-21.

李華仙(1989-),碩士生,研究方向：復(fù)分析與微分方程.

2010 MSC:30D35