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.