Journal of Complex Analysis
Volume 2015 (2015), Article ID 960204, 8 pages
http://dx.doi.org/10.1155/2015/960204

Research Article

Good Linear Operators and Meromorphic Solutions of Functional Equations

Nan Li,1,2 Risto Korhonen,² and Lianzhong Yang¹

¹School of Mathematics, Shandong University, Jinan, Shandong 250100, China
²Department of Physics and Mathematics, Joensuu Campus, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Received 23 March 2015; Accepted 24 April 2015

Academic Editor: Bao Qin Li

Copyright © 2015 Nan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, -difference, and ultradiscrete equations. In many cases, the methodologies used in the study of meromorphic solutions of differential, difference, and -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 , where is a linear polynomial in and , where is good linear operator, is a polynomial in with degree deg , both with small meromorphic coefficients, and is a meromorphic function.

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 -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.

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 say is an exceptional set property if for any two sets and having the property it follows that also has . For instance, “finite linear measure,” “finite logarithmic measure,” and “zero logarithmic density” are exceptional set properties. Denote by the field of meromorphic functions in the complex plane, and let . We say that a linear operator is a good linear operator for with exceptional set property if the following two properties hold:For any ,  as outside of an exceptional set with the property .The counting functions and are asymptotically equivalent; that is, there is a constant such that  as outside of an exceptional set with the property .For example, if and , then property (1) is satisfied by the lemma on the logarithmic derivatives with being “finite linear measure.” Property (2) holds with , even without an error term or an exceptional set. Another example is given by taking to be the set of all meromorphic functions of hyperorder strictly less than one, and . Then property (1) is satisfied by the difference analogue of the lemma on the logarithmic derivatives with being “finite logarithmic measure.” In this case, property (2) holds with .

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. If and are good linear operators for with exceptional set property , then is a good linear operator for with the same exceptional set property .

Proof. Since the linearity follows immediately by the linearity of and , we only need to check that properties (1) and (2) hold for .
First, for any , we have Therefore, since and and by the assumption that and are good operators, we have as outside of a set with exceptional set property . But since as outside of a set with exceptional set property , (4) becomes Thus property (1) holds for the operator .
To show that property (2) also holds, we observe that since , , as outside of a set with exceptional set property , it follows by (5) that Thus property (2) is valid for , and hence it is a good linear operator for with exceptional set property .

As we mentioned in the introduction, the operation of differentiation , , 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 and , and let be the field of meromorphic functions of hyperorder strictly less than one. The operator is a good linear operator in with “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 . If and , 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 .

Lemma 4 (see [10]). Let be a nondecreasing continuous function and let . If the hyperorder of is strictly less than one, that is, and , then where runs to infinity outside of a set of finite logarithmic measure.

Proof of Lemma 2. By Lemma 1 it is sufficient to show that the operators and are good linear operators in with the exceptional set property . The operator is good in fact in all of 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 and for all . By combining (14) with the lemma on the logarithmic derivative, it follows that as outside of a set of finite linear measure. Therefore, if it follows that and thus is a good linear operator in with the exceptional set property .
If , it follows by Lemma 3 that as outside of a set of finite logarithmic measure. Therefore property (1) is satisfied for in with the exceptional set property “finite logarithmic measure.” Moreover, for all , and so, by Lemma 4, we have as outside of a set of finite logarithmic measure. Hence property (2) holds for in with the exceptional set property “finite logarithmic measure.” Finally, by combining (16) and (18), it follows that as outside of a set of finite logarithmic measure. Hence , and so is a good linear operator in with the exceptional set property . 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 where denotes a linear polynomial in and with being a good linear operator, is a polynomial in , and 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 is a linear differential polynomial in with meromorphic coefficients, is a polynomial in with meromorphic coefficients, and is meromorphic, and obtained the following result.

Theorem A. Given , , and as above and , denote by the family of meromorphic solutions of (21) such that whenever , all coefficients of (21) are small meromorphic functions of , and . If now , then Moreover, if , then, for some , for all .

Specialising to , 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 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 .(b)Equation (24) has exactly three transcendental meromorphic solutions , , such that for . Moreover , and for all .

A differential-difference counterpart of Theorems A and B was obtained by Yang and Laine in [12]. They showed that if , is a differential-difference polynomial of , and 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 . 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 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 .

Theorem 5. Let such that, for any , as outside of set with exceptional property , and let be a finite collection of good linear operators for with exceptional set property . If and are any two meromorphic solutions of the equation where is a polynomial in with small meromorphic coefficients, , and is a linear polynomial in and , , with small meromorphic coefficients, then where outside of exceptional set with the property .

The following corollary of Theorem 5 is obtained by choosing as the family of meromorphic functions of hyperorder strictly less than one with relatively few poles and by taking such that , , and .

Corollary 6. Let be a differential-difference polynomial in . If and are any two meromorphic solutions of (27) of hyperorder strictly less than one such that and , then Moreover, if , then, for some , for all . In addition, every meromorphic solution such that hyperorder and satisfies .

Specialising to , 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 , a linear differential-difference polynomial of with small meromorphic coefficients, not vanishing identically, and a meromorphic function. Set . If satisfies the nonlinear differential-difference equation where is a small function of , then one of the following situations holds: (a)Equation (31) has as its unique transcendental meromorphic solution such that .(b)Equation (31) has exactly three transcendental meromorphic solutions , , such that for . Moreover , and for all .

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 be positive integer, let be a small function of , and let denote a linear differential polynomial in : where are small meromorphic functions such that not all are identically zero. Moreover, let 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 .(b)Equation (33) has exactly three transcendental meromorphic solutions , , such that for . Moreover, , and for all .

Following a similar method as in the proof of Theorems 7 and 8, we can generalize the above two results to the case , where .

4. Proof of Theorem 5

By a repeated application of Lemma 1, it follows that, by composing finitely many good linear operators , we obtain another good linear operator for . 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 are good linear operators for with exceptional set property .

Since and are solutions of (27), we have for some . Thus we have where is a polynomial in and with small meromorphic coefficients such that . Now, since the coefficients in (34) are by assumption small with respect to both solutions and , it follows that and for all as outside of a set with exceptional set property . Hence, where and outside of a set with exceptional set property . Therefore, by using Lemma 1 repeatedly and by the definition of the good linear operator, we have where outside of a set with exceptional set property .

Since by assumption (26) we have as outside of a set with exceptional set property , it follows that where again outside of a set with exceptional set property . Therefore (36) becomes an algebraic equation for over the field Therefore, by a generalization of [16, Theorem 1] (see, e.g., [17, p. 34]), it follows that as outside of a set with exceptional set property . This asymptotic equation yields assertion (28).

5. Proof of Corollary 6

As a linear differential-difference polynomial, may be written in the form where and are nonnegative integers, the coefficients are small meromorphic functions with respect to , and are complex constants. By defining and using Lemma 2, (44) becomes where is a good linear operator in for all and 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 Since are small functions for all and and , by using Lemma 2, we have Further since are small functions for all , it follows from the Valiron-Mohon’ko theorem (see, e.g., [17, Theorem 2.2.5]) that Substituting (47) and (48) into (46) and applying [17, Lemma 1.1.1], we get that, for every and , there exists such that provided . Then

On the other hand, writing (27) in the form and by making use of [17, Lemma 1.1.1], (47), and (48), we get that when . Since by the assumption, then from (51) we have Thus .

6. Proof of Theorem 7

Suppose that (31) has at least two distinct transcendental meromorphic solutions such that . Then and are two Borel exceptional values of ; thus is of regular growth. So we have and . From (31), we have where . Here are the distinct third roots of unity. Also, recalling the definition of from (38), we have that as outside of an exceptional set of finite logarithmic measure. Thus by Lemmas 1 and 2 it follows that where outside of an exceptional set of finite logarithmic measure. Moreover, and thus again as outside of an exceptional set of finite logarithmic measure. Assume first that . Then, by the definition of , it follows that for some . Substituting this into (53), we have and so . Thus from (31) we have as asserted. Moreover, and thus for both and . Therefore, , , and are the solutions of (31), and so we have completed the proof of part (b).

Assume now that . Then By (58), we have that where is a polynomial in of degree with constant coefficients. Since , we have as outside of an exceptional set of finite logarithmic measure. Similarly, where, as above, outside of an exceptional set of finite logarithmic measure. Combining (60) and (61), we obtain where the exceptional set is as above. Hence , and so by (60) we have Thus the error terms and are asymptotically equivalent. Assuming that for some , then from (58) we have , or . So we have Since it follows by the second main theorem of Nevanlinna theory that This is a contradiction. Thus (31) may have only one transcendental meromorphic solution such that , as asserted in part (a).

7. Proof of Theorem 8

Suppose that (33) has at least two distinct transcendental meromorphic solutions such that . Then, similarly as in the proof of Theorem 7, where , , and with the error term being defined as in (38) and outside of an exceptional set of finite linear measure. Here are the distinct third roots of unity. Also, we have where outside of an exceptional set of finite linear measure, and by assumption. The case leads to the assertion of part (b) exactly as in the proof of Theorem 7, and so we may assume that . Again, by repeating the reasoning in the proof of Theorem 7, we have that and Assume now that there exists such that . Then . From (67), we have Thus we get So, by (70), (72), and the second main theorem of Nevanlinna theory, we get This is a contradiction. Therefore (33) may have only one transcendental meromorphic solution such that , as asserted in part (a).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the NNSF of China (no. 11171013 and no. 11371225) and the Academy of Finland Grant no. 268009.

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