Lemma 3.1 (see [3, 9]).
Let
be a meromorphic function. Then for all irreducible rational functions in
,
with meromorphic coefficients
and
, the characteristic function of
satisfies
where
and
In the particular case when
we have
Lemma 3.2.
Given distinct complex numbers
a meromorphic function
and meromorphic functions
's, one has
where
. In the particular case when
one has
Remark 3.3.
Observe that the term
does not appear in (3.6). This follows by a careful inspection of the proof of [16, Proposition B.15, Theorem B.16].
Remark 3.4.
Note that the inequality (3.6) remains true, if we replace the characteristic function
by the proximity function
(or by the counting function
).
Lemma 3.5 (see [12, Theorem 2.1]).
Let
be a nonconstant meromorphic function of finite order,
and
. Then
for all
outside of a possible exceptional set
with finite logarithmic measure 
Lemma 3.6 (see [12, Lemma 2.2]).
Let
be a nondecreasing continuous function,
and let
be the set of all
such that
If the logarithmic measure of
is infinite, that is,
then
Proof of Theorem 2.1.
Since the coefficients
's,
and
in (2.2) are small functions relative to
, that is,
hold for all
outside of a possible exceptional set
with finite logarithmic measure 
Let
be a finite order meromorphic solution of (2.2). According to Lemma 3.5, we have, for any
,
where the exceptional set
associated to
is of finite logarithmic measure 
It follows from Lemma 3.6 that
for any
.
Now, equating the Nevanlinna characteristic function on both sides of (2.2), and applying Lemmas 3.1 and 3.2, we have
where
.
Therefore, by (3.13) and (3.14), it follows that
for all
outside of a possible exceptional set
with finite logarithmic measure. Dividing this by
and letting
outside of the exceptional set
and
of
and
, respectively, we have
The proof of Theorem 2.1 is completed.
Example 3.7.
Let
be a constant such that
where
, and let
. We see that
solves
This shows that the equality
is arrived in Theorem 2.1 if 
Example 3.8.
Let
. We see that
solves
This shows that the case
may occur in Theorem 2.1 if 
Lemma 3.9 (see [17]).
Let
be a meromorphic function and let
be given by
Then either
or
Lemma 3.10 (see [15]).
Let
be a nonconstant meromorphic function and let
,
be two polynomials in
with meromorphic coefficients small functions relative to
. If
and
have no common factors of positive degree in
over the field of small functions relative to
, then
Proof of Theorem 2.3.
Suppose that the second alternative of the conclusion is not correct. Then we have, by using Lemmas 3.9, 3.10, 3.2, (2.7), and (2.9),
where 
Thus, we have
Now assuming that
, we have
and for all 
It follows from Lemmas 3.1, 3.2, (3.23), and (2.9) we have
From this, we have
Together with (3.25)–(3.27), we can use method of induction and obtain, for
,
Moreover, we immediately obtain from (3.28) that
and for sufficiently large
, we have
It also follows from Lemma 3.6 that
for any
, assuming that
is of finite order.
Now (3.31) combined with (3.29) and (3.30) yields an immediate contradiction if
. Therefore the only possibility is that
is of infinite order. The proof of Theorem 2.3 is completed.