亞純函數及其n階導數權分擔兩個值

徐洪焱,易才鳳

(1.景德鎮陶瓷學院信息學院,江西景德鎮 333403;2.江西師范大學數信學院,江西南昌 330027)

亞純函數及其n階導數權分擔兩個值

徐洪焱1,易才鳳2

(1.景德鎮陶瓷學院信息學院,江西景德鎮 333403;2.江西師范大學數信學院,江西南昌 330027)

研究亞純函數及其n階導數權分擔兩個值的唯一性問題.得到了:如果兩個非常數亞純函數f,g分擔(∞,∞),f(n)與g(n)分擔(1,0),n(≥0)為一整數,且滿足△C0:= (4n+6)λ+δn+1(0,f)+δn+1(0,g)+δn+2(0,f)+δn+2(0,g)+δn(0,f)＞4n+10,其中λ=max{min{Θ(∞,f),Θ(0,f)},min{Θ(∞,g),Θ(0,g)}},那么f(n)·g(n)≡1,或者f≡g.該結果改進了前人的有關定理.

亞純函數;權分擔;唯一性

1 引言及主要結果

2 引理

3 定理1.1的證明

這樣很容易知道H的極點只可能發生以下幾種情況:(1)F,G的重級零點;(2)F,G的重級極點;(3)F,G的1-值點且其重數不相等;(4)F'的零點但非F(F?1)的零點;(5)G'的零點但非G(G?1)的零點.

(i)由(i)的條件,知F,G分擔(1,0),(∞,∞),再根據H的表達式很容易知道F,G的極點不是H的極點.

如果H/≡0,由引理2.7,有

類似于情形1也可得到矛盾.

如果a?1=0,則f(n)≡g(n).由此方程可得到f=g+p(z),這里p(z)為多項式,顯然T(r,f)=T(r,g)+S(r,f).如果p(z)/≡0,根據引理2.2,有

由(1.1)式很容易得到T(r,f)≤S(r,f),r∈I矛盾.因此p(z)≡0,即f≡g.

綜合三種情形,定理1.1(i)得證.下證定理1.1(ii).

(ii)因為f(n)與g(n)分擔(1,0),所以F,G分擔(1,0).

如果H/≡0,則(3.1)式變為

4 定理1.2的證明

如果H≡0,類似于定理1.1可以證得f(n)·g(n)≡1,或者f≡g,即定理1.2(i)得證.

類似于定理1.1(ii)證明方法并結合(4.1)-(4.4)式及f,g分擔(∞,0),也很容易證明.這里略之.

綜上所述,則完成了定理1.2的證明.

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Mermorphic functions concerning their n-th derivative sharing two values with weight

XU Hong-yan1,YI Cai-feng2

(1.Department of Informatics and Engineering,Jingdezhen Ceramic Institute,Jingdezhen333403,China; 2.Institute of Mathematics and Informatics,Jiangxi Normal University,Nanchang330027,China)

In this paper,we deal with the uniqueness problem of meromorphic functions concerning their n-th derivative sharing two values with weight and obtain the following theorem:if two nonconstant meromorphic functions f,g share(∞,∞),fn,gnshare(1,0),and satisfy△C0:=(4n+6)λ+δn+1(0,f)+δn+1(0,g)+δn+2(0,f)+ δn+2(0,g)＞4n+10,where λ=max{min{Θ(∞,f),Θ(0,f)},min{Θ(∞,g),Θ(0,g)}},then either f(n)·g(n)≡1 or f≡g,where n(＞0)is an integer.These results extend the former theorems.

meromorphic function,weighted sharing,uniqueness

O174.52

A

1008-5513(2009)04-0777-09

2008-04-21.

國家自然科學基金(10871108),景德鎮陶瓷學院科研項目(景陶院發[2009]86號).

徐洪焱(1980-),碩士,研究方向：復分析.

2000MSC:30D30,30D35