Good Linear Operators and Meromorphic Solutions of Functional Equations

Nevanlinna theory provides us withmany tools applicable to the study of value distribution ofmeromorphic solutions of differential equations. Analogues of some of these tools have been recently developed for difference, q-difference, and ultradiscrete equations. In many cases, the methodologies used in the study of meromorphic solutions of differential, difference, and q-difference equations are largely similar.The purpose of this paper is to collect some of these tools in a common toolbox for the study of general classes of functional equations by introducing notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. As an example case, we apply our methods to study the growth of meromorphic solutions of the functional equationM(z, f) + P(z, f) = h(z), whereM(z, f) is a linear polynomial in f and L(f), where L is good linear operator, P(z, f) is a polynomial in f with degree deg P ≥ 2, both with small meromorphic coefficients, and h(z) is a meromorphic function.


Introduction
Lemma on the logarithmic derivatives is an important technical tool in the study of value distribution of meromorphic solutions of differential equations.It is one of the key components in the proof of the Clunie lemma [1] and in a theorem due to A. Z. Mohon'ko and V. D. Mohon'ko [2], both of which are applicable to large classes of differential equations.Similarly, the difference analogues of the lemma on the logarithmic derivatives due to Halburd and the second author [3,4] and Chiang and Feng [5,6] are applicable to study large classes of difference equations, often by using methods similar to the case of differential equations.A -difference analogue [7] of the lemma on the logarithmic derivatives, as well as an analogous result on the proximity function of polynomial compositions of meromorphic functions [8], is applicable to corresponding classes of -difference equations and functional equations much in the same way.Therefore it is natural to present all these results under one general framework.For value distribution of meromorphic functions, this was done in [9], where a second main theorem was given for general linear operators, operating on a subfield of meromorphic functions for which a suitable analogue of the lemma on the logarithmic derivative exists.The purpose of this paper is to develop this method further so that it is applicable to equations and to apply it to study meromorphic solutions of a general class of functional equations.This will be done in Section 2 by introducing the notion of a good linear operator, which encompasses such operators as () =   ,   () = (), and   () = ( + ).In Section 3 we apply our methods to study the existence and uniqueness and the growth of meromorphic solutions of a general class of functional equations.Sections 4-7 contain the proofs of the results stated in Section 3.

Good Linear Operators
The lemma on the logarithmic derivative and its difference analogues all produce different types of exceptional sets.In order to include this phenomenon in our setup, we introduce the following notion.We say P is an exceptional set property if for any two sets  1 ⊂ (0, ∞) and  2 ⊂ (0, ∞) having the property P it follows that  1 ∪  2 also has P.For instance, "finite linear measure, " "finite logarithmic measure, " and "zero logarithmic density" are exceptional set properties.Denote by M the field of meromorphic functions in the complex plane, and let N ⊂ M. We say that a linear operator  : N → N is a good linear operator for N with exceptional set property P if the following two properties hold: (1) For any  ∈ N, as  → ∞ outside of an exceptional set   with the property P.
(2) The counting functions (, ) and (, ()) are asymptotically equivalent; that is, there is a constant  ≥ 1 such that as  → ∞ outside of an exceptional set   with the property P.
For example, if N = M and () =   , then property (1) is satisfied by the lemma on the logarithmic derivatives with P being "finite linear measure." Property (2) holds with  = 2, even without an error term or an exceptional set.Another example is given by taking N to be the set of all meromorphic functions of hyperorder strictly less than one, and (()) = ( + 1).Then property (1) is satisfied by the difference analogue of the lemma on the logarithmic derivatives with P being "finite logarithmic measure." In this case, property (2) holds with  = 1.
The following result shows that a composition of two good operators is also a good operator.Note, however, that the sum of two good linear operators is not necessarily a good operator, since the lower bound in (2) may fail to be valid.
as  → ∞ outside of a set with exceptional set property P, (4) becomes Thus property (1) holds for the operator  1 ∘  2 .
To show that property (2) also holds, we observe that since as  → ∞ outside of a set with exceptional set property P, it follows by ( 5) that Thus property ( 2) is valid for  1 ∘  2 , and hence it is a good linear operator for N with exceptional set property P.
As we mentioned in the introduction, the operation of differentiation  : M → M, () =   , is a good linear operator with the exceptional set property "finite linear measure." Lemma 1 implies that a composition of single term differential and difference operators of arbitrary order is a good linear operator for sufficiently slowly growing meromorphic functions.

Lemma 2.
Let  ∈ C and  ∈ N ∪ {0}, and let N 1 be the field of meromorphic functions of hyperorder strictly less than one.The operator is a good linear operator in N 1 with P = "finite logarithmic measure." In order to prove this lemma, we need the following two results from the field of difference Nevanlinna theory.The first is a difference analogue of the lemma on the logarithmic derivatives.
Lemma 3 (see [10]).Let () be a nonconstant meromorphic function and and  > 0, then for all  outside of a set of finite logarithmic measure.
The second auxiliary lemma helps us to deal with shifted counting functions in the field N 1 .
Proof of Lemma 2. By Lemma 1 it is sufficient to show that the operators  1 () =   and  2 (()) = ( + ) are good linear operators in N 1 with the exceptional set property P. The operator  1 is good in fact in all of M with a weaker exceptional set property.Namely, property (1) is satisfied by the lemma on the logarithmic derivative, and property (2) holds since for all meromorphic functions  ∈ M and for all  ≥ 1.By combining (14) with the lemma on the logarithmic derivative, it follows that  (, as  → ∞ outside of a set of finite logarithmic measure.Hence  2 (N 1 ) ⊂ N 1 , and so  2 : N 1 → N 1 is a good linear operator in N 1 with the exceptional set property P.This completes the proof of Lemma 2.

Meromorphic Solutions of a Functional Equation
In this section we apply the concept of good linear operator to study meromorphic solutions of where (, ) denotes a linear polynomial in  and () with  being a good linear operator, (, ) is a polynomial in , and ℎ() is a meromorphic function.Equation ( 20) is an extension of a differential equation studied by Heittokangas et al. [11] in 2002.They considered the growth of meromorphic solutions of where () =  0 () +  1 ()  + ⋅ ⋅ ⋅ +   () () is a linear differential polynomial in  with meromorphic coefficients, (, ) =  2 () 2 + ⋅ ⋅ ⋅ +   ()  is a polynomial in  with meromorphic coefficients, and ℎ() is meromorphic, and obtained the following result.
Specialising to ()−() 3 = ℎ(), where () is a small meromorphic function, Heittokangas et al. [11] also considered the existence and uniqueness of meromorphic solutions with few poles only and obtained the following result.

Theorem B. Let 𝑓 be a transcendental meromorphic function.
If  satisfies the nonlinear differential equation then one of the following situations hold: (a) Equation ( 24) has  as its unique transcendental meromorphic solution such that (, ) = (, ).
A differential-difference counterpart of Theorems A and B was obtained by Yang and Laine in [12].They showed that if  ≥ 4, (, ) ̸ ≡ 0 is a differential-difference polynomial of , and ℎ is a meromorphic function of finite order, then the equation possesses at most one admissible transcendental entire solution of finite order and that if such a solution exists, it is of the same order as ℎ.Further results on difference and differentialdifference related to (25) can be found, for example, in [13][14][15].
In the following theorem we apply the concept of good linear operator introduced in Section 2 to obtain a natural extension of Theorem A and of its difference analogue to a general class of functional equations.In order to state our generalization, we say that meromorphic function  is small with respect to  if (, ) = ((, )) as  → ∞ outside of an exceptional set with the exceptional set property P.
where  → ∞ outside of exceptional set  with the property P.
Specialising to (, ) − () 3 = ℎ(), where () is a small meromorphic function, we obtain the following result on the existence of meromorphic solutions.Theorem 7. Let  be an transcendental meromorphic function of hyperorder  2 < 1, (, ) a linear differential-difference polynomial of  with small meromorphic coefficients, not vanishing identically, and ℎ a meromorphic function.Set   = max{(), (1/)}.If  satisfies the nonlinear differentialdifference equation where () ( ̸ ≡ 0) is a small function of , then one of the following situations holds: If, in particular, we restrict the linear differential-difference polynomial (, ) to be linear differential polynomial (, ), then we get the following result which improves Theorem B. Theorem 8. Let  be a transcendental meromorphic function such that (, ) = (, ).Moreover, let  (≥ 1) be positive integer, let () ( ̸ ≡ 0) be a small function of , and let () denote a linear differential polynomial in : where   () ( = 0, . . ., ) are small meromorphic functions such that not all   are identically zero.Moreover, let ℎ() be a meromorphic function.If  is a solution of the nonlinear differential equation then one of the following situations hold: (a) Equation ( 33) has  as its unique transcendental meromorphic solution such that (, ) = (, ).
Following a similar method as in the proof of Theorems 7 and 8, we can generalize the above two results to the case (, ) = −()  (), where  ≥ 3.

Proof of Theorem 5
By a repeated application of Lemma 1, it follows that, by composing finitely many good linear operators N, we obtain another good linear operator for N. Since (, ) is a linear polynomial in  and in the good linear operators   (), where  ∈ , it follows that (, ) can be written, without loss of generality, in the form where the coefficients   are small meromorphic functions with respect to  and L1 , . . ., L are good linear operators for N with exceptional set property P.
Since  1 and  2 are solutions of (27), we have for some  ≥ 2. Thus we have where (, as  → ∞ outside of a set with exceptional set property P.This asymptotic equation yields assertion (28).

Proof of Corollary 6
As a linear differential-difference polynomial, (, ) may be written in the form where  and ℓ are nonnegative integers, the coefficients   () are small meromorphic functions with respect to , and  0 , . . .,   are complex constants.

Lemma 1. If 𝐿 1 and 𝐿 2 are good linear operators for N with exceptional set property P, then 𝐿 1 ∘𝐿 2 is a good linear operator
for N with the same exceptional set property P.