# Good Linear Operators and Meromorphic Solutions of Functional Equations

Authors: Nan Li; Risto Korhonen; Lianzhong Yang
Journal: Journal of Complex Analysis
 (2015)

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## Abstract

Nevanlinna theory provides us with many tools applicable to the study of value distribution of meromorphic 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 equation M(z,f)+P(z,f)=h(z), where M(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.


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## Body

## 1. 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 q-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 q-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 L(f)=f′, Lq(f)=f(qz), and Ec(f)=f(z+c). 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.


## 2. 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 sayP is anexceptional set property if for any two sets E1⊂(0,∞) and E2⊂(0,∞) having the property P it follows that E1∪E2 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 L:N→N is agood linear operator forNwith exceptional set propertyP if the following two properties hold:(1)
For anyf∈N, (1)mr,Lff=oTr,fasr→∞ outside of an exceptional set Ef with the property P.(2)
The counting functionsN(r,f) and N(r,L(f)) are asymptotically equivalent; that is, there is a constant K≥1 such that (2)1KNr,f+oTr,f≤Nr,Lf≤KNr,f+oTr,fasr→∞ outside of an exceptional set Ef with the property P.For example, if N=M and L(f)=f′, then property (1) is satisfied by the lemma on the logarithmic derivatives with P being “finite linear measure.” Property (2) holds with K=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 L(f(z))=f(z+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 K=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.Lemma 1.
IfL1 and L2 are good linear operators for N with exceptional set property P, then L1∘L2 is a good linear operator for N with the same exceptional set property P.Proof.
Since the linearity follows immediately by the linearity ofL1 and L2, we only need to check that properties (1) and (2) hold for L1∘L2.
First, for anyf∈N, we have (3)mr,L1L2ff≤mr,L1L2fL2f+mr,L2ff.Therefore, since f∈N and L2(f)∈N and by the assumption that L1 and L2 are good operators, we have (4)mr,L1L2ff=oTr,L2f+oTr,fas r→∞ outside of a set with exceptional set property P. But since (5)Tr,L2f=mr,L2f+Nr,L2f≤mr,f+mr,L2ff+KNr,f+oTr,f≤KTr,f+oTr,fas r→∞ outside of a set with exceptional set property P, (4) becomes (6)mr,L1L2ff=oTr,f.Thus property (1) holds for the operator L1∘L2.
To show that property (2) also holds, we observe that sinceL2(f)∈N, f∈N, (7)1K1Nr,L2f+oTr,L2f≤Nr,L1L2f≤K1Nr,L2f+oTr,L2f,1K2Nr,f+oTr,f≤Nr,L2f≤K2Nr,f+oTr,fas r→∞ outside of a set with exceptional set property P, it follows by (5) that (8)1K1K2Nr,f+oTr,f≤Nr,L1L2f≤K1K2Nr,f+oTr,f.Thus property (2) is valid for L1∘L2, and hence it is a good linear operator for N with exceptional set property P.As we mentioned in the introduction, the operation of differentiationL:M→M, L(f)=f′, 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.
Letc∈C and k∈N∪{0}, and let N1 be the field of meromorphic functions of hyperorder strictly less than one. The operator (9)Lfz=fkz+cis a good linear operator in N1 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]).
Letf(z) be a nonconstant meromorphic function and c∈C. If (10)σ2f≔lim⁡sup⁡r→∞⁡log⁡⁡log⁡⁡Tr,flog⁡⁡r=σ2<1and ε>0, then (11)mr,fz+cfz=oTr,fr1-σ2-εfor all r outside of a set of finite logarithmic measure.The second auxiliary lemma helps us to deal with shifted counting functions in the fieldN1.Lemma 4 (see [10]).
LetT:[0,+∞)→[0,+∞) be a nondecreasing continuous function and let s∈(0,∞). If the hyperorder of T is strictly less than one, that is, (12)lim⁡sup⁡r→∞⁡log⁡⁡log⁡⁡Trlog⁡⁡r=σ2<1and δ∈(0,1-σ2), then (13)Tr+s=Tr+oTrrδ,where r runs to infinity outside of a set of finite logarithmic measure.Proof of Lemma2.
By Lemma1 it is sufficient to show that the operators L1(f)=f′ and L2(f(z))=f(z+c) are good linear operators in N1 with the exceptional set property P. The operator L1 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 (14)Nr,f≤Nr,L1f≤2Nr,ffor all meromorphic functions f∈M and for all r≥1. By combining (14) with the lemma on the logarithmic derivative, it follows that (15)Tr,L1f≤2Tr,f+oTr,fas r→∞ outside of a set of finite linear measure. Therefore, if f∈N1 it follows that L1(f)∈N1 and thus L1:N1→N1 is a good linear operator in N1 with the exceptional set property P.
Iff∈N1, it follows by Lemma 3 that (16)mr,L2ff=mr,fz+cfz=oTr,fas r→∞ outside of a set of finite logarithmic measure. Therefore property (1) is satisfied for L2 in N1 with the exceptional set property “finite logarithmic measure.” Moreover, (17)Nr-c,fz≤Nr,L2f≤Nr+c,fzfor all r≥|c|, and so, by Lemma 4, we have (18)Nr,fz+oNr,f≤Nr,L2f≤Nr,fz+oNr,fas r→∞ outside of a set of finite logarithmic measure. Hence property (2) holds for L2 in N1 with the exceptional set property “finite logarithmic measure.” Finally, by combining (16) and (18), it follows that (19)Tr,L2f≤Tr,f+oTr,fas r→∞ outside of a set of finite logarithmic measure. Hence L2(N1)⊂N1, and so L2:N1→N1 is a good linear operator in N1 with the exceptional set property P. This completes the proof of Lemma 2.


## 3. Meromorphic Solutions of a Functional Equation

In this section we apply the concept of good linear operator to study meromorphic solutions of(20)Mz,f+Pz,f=hz,where M(z,f) denotes a linear polynomial in f and L(f) with L being a good linear operator, P(z,f) is a polynomial in f, and h(z) 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 (21)Lf+Pz,f=hz,where Lf=a0zf+a1zf′+⋯+ak(z)f(k) is a linear differential polynomial in f with meromorphic coefficients, Pz,f=b2zf2+⋯+bn(z)fn is a polynomial in f with meromorphic coefficients, and h(z) is meromorphic, and obtained the following result.Theorem A.
GivenL(f), P(z,f), and h(z) as above and P(z,f)≢0, denote by F the family of meromorphic solutions of (21) such that whenever f∈F, all coefficients of (21) are small meromorphic functions of f, and N(r,f)=S(r,f). If now f,g∈F, then (22)Tr,g=OTr,f+Sr,f.Moreover, if α>1, then, for some rα>0, (23)Tr,g=OTαr,ffor all r≥rα.Specialising toL(f)-p(z)f3=h(z), where p(z) 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.
Letf be a transcendental meromorphic function. If f satisfies the nonlinear differential equation (24)a1zf′+a0zf-pzf3=hz,then one of the following situations hold: (a)
Equation (24) has f as its unique transcendental meromorphic solution such that N(r,f)=S(r,f).(b)
Equation (24) has exactly three transcendental meromorphic solutions fj, j=1,2,3, such that N(r,fj)=S(r,fj) for j=1,2,3. Moreover a1(z)fj′+a0(z)fj≡0, and h(z)=-p(z)fj3 for all j=1,2,3.A differential-difference counterpart of Theorems A and B was obtained by Yang and Laine in [12]. They showed that if n≥4, M(z,f)≢0 is a differential-difference polynomial of f, and h is a meromorphic function of finite order, then the equation (25)fzn+Mz,f=hzpossesses 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 differential-difference related to (25) can be found, for example, in [13–15].In the following theorem we apply the concept of good linear operator introduced in Section2 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 g is small with respect to f if T(r,g)=o(T(r,f)) as r→∞ outside of an exceptional set with the exceptional set property P.Theorem 5.
LetN⊂M such that, for any f∈N, (26)Nr,f=oTr,fas r→∞ outside of set E with exceptional property P, and let Lk:k∈J be a finite collection of good linear operators for N with exceptional set property P. If f1∈N and f2∈N are any two meromorphic solutions of the equation (27)Mz,f+Pz,f=hz,where P(z,f)=b2(z)f2+⋯+bn(z)fn is a polynomial in f with small meromorphic coefficients, h∈M, and M(z,f) is a linear polynomial in f and Lk(f), k∈J, with small meromorphic coefficients, then (28)Tr,f2=OTr,f1+oTr,f1,where r→∞ outside of exceptional set E with the property P.The following corollary of Theorem5 is obtained by choosing N as the family of meromorphic functions of hyperorder strictly less than one with relatively few poles and by taking L1,…Ll+1 such that Lk(f)=f(z+ck), k=1,…,l, and Ll+1(f)=f′.Corollary 6.
LetM(z,f) be a differential-difference polynomial in f. If f1 and f2 are any two meromorphic solutions of (27) of hyperorder strictly less than one such that N(r,f1)=S(r,f1) and N(r,f2)=S(r,f2), then (29)Tr,f2=OTr,f1+Sr,f1.Moreover, if α>1, then, for some rα>0, (30)Tr,f2=OTαr,f1for all r≥rα. In addition, every meromorphic solution such that hyperorder σ2<1 and N(r,f)=S(r,f) satisfies ρ(f)=ρ(h).Specialising toM(z,f)-p(z)f3=h(z), where p(z) is a small meromorphic function, we obtain the following result on the existence of meromorphic solutions.Theorem 7.
Letf be an transcendental meromorphic function of hyperorder σ2<1, M(z,f) a linear differential-difference polynomial of f with small meromorphic coefficients, not vanishing identically, and h a meromorphic function. Set λf=max⁡{λ(f),λ(1/f)}. If f satisfies the nonlinear differential-difference equation (31)Mz,f-pzfz3=hz,where p(z)(≢0) is a small function of f, then one of the following situations holds: (a)
Equation (31) has f as its unique transcendental meromorphic solution such that λf<σf.(b)
Equation (31) has exactly three transcendental meromorphic solutions fj, j=1,2,3, such that λfj<σfj for j=1,2,3. Moreover M(z,fj)≡0, and h(z)=-p(z)fj3 for all j=1,2,3.If, in particular, we restrict the linear differential-difference polynomialM(z,f) to be linear differential polynomial L(z,f), then we get the following result which improves Theorem B.Theorem 8.
Letf be a transcendental meromorphic function such that N(r,f)=S(r,f). Moreover, let k(≥1) be positive integer, let p(z)(≢0) be a small function of f, and let L(f) denote a linear differential polynomial in f: (32)Lf=a0zf+a1zf′+⋯+akzfk,where aj(z)(j=0,…,k) are small meromorphic functions such that not all aj are identically zero. Moreover, let h(z) be a meromorphic function. If f is a solution of the nonlinear differential equation (33)Lf-pzfz3=hz,then one of the following situations hold: (a)
Equation (33) has f as its unique transcendental meromorphic solution such that N(r,f)=S(r,f).(b)
Equation (33) has exactly three transcendental meromorphic solutions fj, j=1,2,3, such that N(r,fj)=S(r,fj) for j=1,2,3. Moreover, L(fj)≡0, and h(z)=-p(z)fj3 for all j=1,2,3.Following a similar method as in the proof of Theorems7 and 8, we can generalize the above two results to the case P(z,f)=-p(z)fn(z), where n≥3.


## 4. Proof of Theorem5

By a repeated application of Lemma1, it follows that, by composing finitely many good linear operators N, we obtain another good linear operator for N. Since M(z,f) is a linear polynomial in f and in the good linear operators Lj(f), where j∈J, it follows that M(z,f) can be written, without loss of generality, in the form (34)Mz,f=∑j=1pajzL~jfz,where the coefficients aj are small meromorphic functions with respect to f and L~1,…,L~p are good linear operators for N with exceptional set property P.Sincef1 and f2 are solutions of (27), we have (35)Mz,f1-Mz,f2f1-f2=-Pz,f1-Pz,f2f1-f2=-bnf1n-f2n+⋯+b2f12-f22f1-f2for some n≥2. Thus we have (36)Mz,f1-f2f1-f2=Mz,f1-Mz,f2f1-f2=Qz,f1,f2,where Q(z,f1,f2) is a polynomial in f1 and f2 with small meromorphic coefficients such that deg⁡f1Q=deg⁡f2Q≥1. Now, since the coefficients aj in (34) are by assumption small with respect to both solutions f1 and f2, it follows that T(r,aj)=o(T(r,f1)) and T(r,aj)=o(T(r,f2)) for all j∈1,…,p as r→∞ outside of a set with exceptional set property P. Hence, (37)Tr,aj=oTr,j∈1,…,p,where (38)Tr≔max⁡⁡Tr,f1,Tr,f2and r→∞ outside of a set with exceptional set property P. Therefore, by using Lemma 1 repeatedly and by the definition of the good linear operator, we have (39)mr,Qz,f1,f2=mr,Mz,f1-f2f1-f2≤mr,∑j=1pajzL~jf1-f2f1-f2=oTr,f1-f2+oTr=oTr,where r→∞ outside of a set with exceptional set property P.Since by assumption (26) we have (40)max⁡⁡Nr,f1,Nr,f2=oTras r→∞ outside of a set with exceptional set property P, it follows that (41)Tr,Qz,f1,f2=oTr,where again r→∞ outside of a set with exceptional set property P. Therefore (36) becomes an algebraic equation for f2 over the field (42)L≔hmeromorphic:Tr,h=OTr,f1+oTr.Therefore, by a generalization of [16, Theorem 1] (see, e.g., [17, p. 34]), it follows that (43)Tr,f2=OTr,f1+oTras r→∞ outside of a set with exceptional set property P. This asymptotic equation yields assertion (28).


## 5. Proof of Corollary6

As a linear differential-difference polynomial,M(z,f) may be written in the form (44)Mz,f=∑i=0k∑j=0laijzfjz+ci,where k and l are nonnegative integers, the coefficients aij(z) are small meromorphic functions with respect to f, and c0,…,ck are complex constants. By defining Lij(f(z))=f(j)(z+ci) and using Lemma 2, (44) becomes (45)Mz,f=∑i=0k∑j=0laijzLijfz,where Lij is a good linear operator in N1 for all i and j with the exceptional set property of “finite logarithmic measure.” Hence Theorem 5 implies (29). By applying [17, Lemma 1.1.1] to remove the exceptional set, we get (30).On the other hand, from (27), we have (46)Tr,h≤Tr,Mz,f+Tr,Pz,f+O1.Since aij are small functions for all i=0,…,k and j=0,…,l and N(r,f)=S(r,f), by using Lemma 2, we have (47)Tr,Mz,f=mr,Mz,f+Nr,Mz,f≤mr,Mz,ff+mr,f+Sr,f≤∑i=0k∑j=0lmr,aij·Lijff+mr,f+Sr,f=Tr,f+Sr,f.Further since bj are small functions for all j=2,…,n, it follows from the Valiron-Mohon’ko theorem (see, e.g., [17, Theorem 2.2.5]) that (48)Tr,Pz,f=nTr,f+Sr,f.Substituting (47) and (48) into (46) and applying [17, Lemma 1.1.1], we get that, for every ε>0 and α>1, there exists rα>0 such that (49)Tr,h≤n+1+εTαr,f,provided r≥rα. Then (50)ρh≤lim⁡sup⁡r→∞⁡log⁡⁡Tαr,f+O1log⁡⁡r=lim⁡sup⁡r→∞⁡log⁡⁡Tαr,flog⁡⁡αr-log⁡⁡α=ρf.On the other hand, writing (27) in the form P(z,f)=-M(z,f)+h(z) and by making use of [17, Lemma 1.1.1], (47), and (48), we get that (51)n-1-εTr,f≤Tαr,hwhen r≥rα. Since n≥2 by the assumption, then from (51) we have (52)ρf≤lim⁡sup⁡r→∞⁡log⁡⁡Tαr,hlog⁡⁡αr-log⁡⁡α=ρh.Thus ρ(f)=ρ(h).


## 6. Proof of Theorem7

Suppose that (31) has at least two distinct transcendental meromorphic solutions fi(i=1,2) such that λfi<σfi. Then 0 and ∞ are two Borel exceptional values of fi; thus fi is of regular growth. So we have N(r,fi)=S(r,fi) and N(r,1/fi)=S(r,fi). From (31), we have (53)Mz,f1-f2f1-f2=Mz,f1-Mz,f2f1-f2=pzf13-f23f1-f2=pzFz,where F(z)=(f1-ε1f2)(f1-ε2f2). Here εj≠1 are the distinct third roots of unity. Also, recalling the definition of T(r) from (38), we have that T(r,p)=o(T(r)) as r→∞ outside of an exceptional set of finite logarithmic measure. Thus by Lemmas 1 and 2 it follows that (54)mr,F=mr,1p·Mz,f1-f2f1-f2=Sr,f1-f2+oTr=oTr,where r→∞ outside of an exceptional set of finite logarithmic measure. Moreover, (55)Nr,F=oTr,and thus (56)Tr,F=oTragain as r→∞ outside of an exceptional set of finite logarithmic measure. Assume first that F≡0. Then, by the definition of F, it follows that f2=εjf1 for some j=1,2. Substituting this into (53), we have (57)Mz,f1-εjf1=1-εjMz,f1=0,and so M(z,f1)=0. Thus from (31) we have h(z)=-p(z)f13(z) as asserted. Moreover, M(z,εjf1)=0 and thus h(z)=-p(z)(εjf1(z))3 for both j=1 and j=2. Therefore, f1, ε1f1, and ε2f1 are the solutions of (31), and so we have completed the proof of part (b).Assume now thatF≢0. Then (58)Fz=f121-ε1f2f11-ε2f2f1=ε1ε2f12ε12-f2f1ε22-f2f1.By (58), we have that (59)Fzf12=Qf2f1,where Q is a polynomial in f2/f1 of degree 2 with constant coefficients. Since T(r,F)=o(T(r)), we have (60)2Tr,f2f1+O1=Tr,Ff12=2Tr,f1+oTras r→∞ outside of an exceptional set of finite logarithmic measure. Similarly, (61)2Tr,f2f1+O1=2Tr,f1f2+O1=Tr,Ff22=2Tr,f2+oTr,where, as above, r→∞ outside of an exceptional set of finite logarithmic measure. Combining (60) and (61), we obtain (62)Tr,f1=Tr,f2+oTr,where the exceptional set is as above. Hence S(r,f1)=S(r,f2), and so by (60) we have (63)Tr,f2f1=Tr,f1+Sr,f1.Thus the error terms o(T(r)) and Sr,f2/f1 are asymptotically equivalent. Assuming that (f2/f1)(z0)=εj2 for some j=1,2, then from (58) we have F(z0)=0, or f1(z0)=∞. So we have (64)N¯r,1f2/f1-εj2=Sr,f2f1,j=1,2.Since (65)N¯r,1f2/f1=N¯r,f1f2≤N¯r,f1+N¯r,1f2=Sr,f2f1,it follows by the second main theorem of Nevanlinna theory that (66)Tr,f2f1≤N¯r,1f2/f1+N¯r,1f2/f1-ε12+N¯r,1f2/f1-ε22=Sr,f2f1.This is a contradiction. Thus (31) may have only one transcendental meromorphic solution such that λf<σf, as asserted in part (a).


## 7. Proof of Theorem8

Suppose that (33) has at least two distinct transcendental meromorphic solutions fi(i=1,2) such that N(r,fi)=S(r,fi). Then, similarly as in the proof of Theorem 7, (67)∑i=0kaizf1-f2if1-f2=pzf13-f23f1-f2=pzFz,where F(z)=(f1-ε1f2)(f1-ε2f2), T(r,ai)=o(T(r)), and T(r,p)=o(T(r)) with the error term o(T(r)) being defined as in (38) and r→∞ outside of an exceptional set of finite linear measure. Here εj≠1 are the distinct third roots of unity. Also, we have (68)mr,F=mr,1p·∑i=0kaizf1-f2if1-f2=Sr,f1-f2+oTr=oTr,where r→∞ outside of an exceptional set of finite linear measure, and (69)Nr,F=oTrby assumption. The case F=0 leads to the assertion of part (b) exactly as in the proof of Theorem 7, and so we may assume that F≢0. Again, by repeating the reasoning in the proof of Theorem 7, we have that T(r,F)=Sr,f2/f1 and (70)N¯r,1f2/f1-εj2=Sr,f2f1,j=1,2.Assume now that there exists z0∈C such that (f2/f1)(z0)=1. Then (f1-f2)(z0)=0. From (67), we have (71)N¯1f1-f2≤∑i=0kNr,aiz+Nr,1aiz+Nr,pz+Nr,Fz+Sr,f2f1=Sr,f2f1.Thus we get (72)N¯r,1f2/f1-1=Sr,f2f1.So, by (70), (72), and the second main theorem of Nevanlinna theory, we get (73)Tr,f2f1≤N¯r,1f2/f1-1+N¯r,1f2/f1-ε12+N¯r,1f2/f1-ε22=Sr,f2f1.This is a contradiction. Therefore (33) may have only one transcendental meromorphic solution such that N(r,f)=S(r,f), as asserted in part (a).



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