A theorem of meromorphic functions

Zhou Houqing

(Department of math, shaoyang college, hunan, 422000)

Abstract In this paper, we proved the following theorem:

Let andbe two distinct nonconstant meromorphic functions such that andshare 0, 1CM, and, .Let be a complex number and define . If are linearly independent, then

 

 

1,Main results

We say that two meromorphic functions and share a values CM (Counting Multiplicities), if and have the same points(or the same poles when =)with the same multiplicities(see[1]).Throughout this paper we shall employ the standard notions of Nevanlinna theory, such as for where E denotes a subset in with finite measure (see[2]).

In 1980, H.Ueda[3]proved the following

Theorem A Letand be two distinct nonconstant entire functions such that and share 0,1CM,and be a complex number. If is lacunary for ,then is lacunary for ,and

In 1988,Yi[4] improved the above result by proving

Theorem B Letand be two distinct nonconstant entire functions such that and share 0,1CM,and be a complex number. If ,then and are Picard exceptional values of and resp., and the result of theorem A holds.

As to for the case when and are meromorphic functions, Ye[5] in 1992 gave

Theorem C Letand be two distinct nonconstant meromorphic functions such that and share 0,1 and CM, Let, , be n distinct complex numbers. such thatIf , then there exists one and only one in ,such that and are Picard exceptional values of and resp,and so is ,and

In this paper we shall improve the theorem:

Let andbe two distinct nonconstant meromorphic functions such that andshare 0, 1CM, and, .Let be a complex number and define . If are linearly independent, then

(1.1)

(1.2)

 

2. Lemmas

Lemmas1[6] Let be linearly independent meromorphic functions satisfyingThen for We have as outside a set E of finite measure, where D denotes the Wronskinan and denotes the maximum of ,

By using nevanlinnas second fundamental theorem, we can easily prove lemma 2 and lemma 3 as follows.

Lemma 2 Let and be two nonconstant meromorphic functions, and be nonzero constants. If , then

Lemma 3 Letandbe two nonconstant meromorphic functions, If and share 0,1CM,and as outside a set E of finite measure.

Lemma 4[7] Let be meromorphic functions satisfying Let , then are linearly independentSo are

 

3.Proof of theorem

Proof : put (3.1)

By the assumption, it is clear that Also, we have


By lemma3 , Thus we have

(3.2)

Since , and by (3.2), we obtain

(3.3)

Similarly, since we have

(3.4)

From the definitions of and (3.1), we have

and (3.5)

Therefore, by using lemma 1,we obtain

(3.6)

Where , and

And by the definitions of and lemma 3, we have

(3.7)

Also

Hence, by(3.2),(3.6),(3.7),we derive

(3.8)

Thus (3.3) and (3.8) yield (1.1).

To get (1.2), we put

Then , and by lemma 4, are also linearly independent. So similarly as we derived(3.8), we obtain

Which,combining(3.4),gives(1.2).

 

 

 

References

1.       Gundersen G G. Meromorphic functions that share three or four values. J London Math Soc ,1979,20:457~466

2.       hayman W K. Meromorphic functions.ClarendonPress, Oxford,1964

3.       Ueda H. Unicity theorems for meromorphic or entire functions.Kodai Math J,1980,3:457~471

4.       Yi Hong Xun. Meromorphic functions that share three values.Chin Ann Math,1988,9A: 434~439

5.       Ye Shou Zhen. Uniqueness of meromorphic functions that share three values. Kodai ,Math J,1992,15:236~243

6.       Gross F. Factorization of meromorphic functions.D C: U S Govt Printing Office,Washington D.C.,1972

7.       Yi Hong Xun. Meromorphic functions that share two or three values. Kosai Math J,1990,13:363~372