_{}A theorem of meromorphic functions

(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).

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