A theorem of meromorphic functions
Zhou Houqing
(Department of math, shaoyang college, hunan, 422000)
Abstract In this paper, we proved the following theorem:
Let 
and
be two distinct nonconstant meromorphic functions such that 
and
share 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
Let
and 
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
Let
and 
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
Let
and 
be two distinct nonconstant
meromorphic functions such that 
and 
share 0,1 and 
CM, Let
, …, 
be n
distinct complex numbers. such that
。If 
＞
, 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 
and
be two distinct nonconstant meromorphic functions such that 
and
share 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 satisfying
Then for 
We have 
as 
outside a set E of
finite measure, where D denotes the
Wronskinan 
and 
denotes the maximum
of 
,
By using nevanlinna
s 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
Let
and
be 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 independent
So 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