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

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