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
In the particular case when
Given distinct complex numbers a meromorphic function and meromorphic functions 's, one has
where . In the particular case when
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].
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
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.
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
Let . We see that solves
This shows that the case may occur in Theorem 2.1 if
Lemma 3.9 (see ).
Let be a meromorphic function and let be given by
Lemma 3.10 (see ).
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),
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.