On the Complex Oscillation of Meromorphic Solutions of Nonhomogeneous Linear Differential Equations with Meromorphic Coefficients

Abstract

In this paper, we study the higher order differential equation , where is a rational function, having a pole at of order , and is a meromorphic function with finite order, and obtain some properties related to the order and zeros of its meromorphic solutions.

1. Introduction and Results

In this paper, a meromorphic function means a function that is meromorphic in the whole complex plane . We assume that the reader is familiar with the fundamental results and the standard notations of Nevanlinna’s value distribution theory of meromorphic functions (see [1, 2]) and use the same notations as in [3]. In addition, we use the notations and to denote the exponents of convergence of the zeros and distinct zeros of a meromorphic function , respectively, and to denote the growth order of .

In 1993, Chen [4] obtained the following theorems.

Theorem 1. Let be a rational function with -th order pole at , and let be a transcendental meromorphic function with . If all solutions of the differential equationare meromorphic, then every solution of (1) satisfieswith at most one exceptional solution satisfying .

Theorem 2. Let be a rational function with -th order pole at , and let be a transcendental meromorphic function with . If all solutions of (1) are meromorphic, then(a).(b)If , then .(c)If , then satisfieswith at most one exceptional solution satisfying .

We all know from the fundamental theory of complex differential equation that all solutions of linear differential equation with entire coefficients are entire functions, but a solution of linear differential equation with meromorphic coefficients is not always a meromorphic function. For example, is a solution of the equation , but is not a meromorphic function. Hence, the condition “all solutions of (1) are meromorphic” in Theorems 1 and 2 is very rigorous. A natural question to ask is whether the condition “all solutions of (1) are meromorphic” in Theorems 1 and 2 can be omitted? In this paper, we consider the above problem and give a positive answer by proving the following theorems.

Theorem 3. Let be a rational function with -th order pole at , and let be a meromorphic function with . If the differential equationhas a meromorphic solution, then(a)If , then all meromorphic solutions of (4) satisfy with at most one exceptional solution satisfying .(b) If , then all meromorphic solutions of (4) satisfy and

Theorem 4. Let be a rational function with -th order pole at , and let be a meromorphic function with . If differential equation (4) has meromorphic solution, then(a).(b)If , then .(c)If , then all meromorphic solutions of (4) satisfywith at most one exceptional solution satisfying

2. Lemmas for the Proof of Theorems

Lemma 1 (see [5]). Let be a meromorphic function and . Then, for any , there exists a set that has finite linear measure and finite logarithmic measure, such thatholds for all satisfying .

Lemma 2 (see [5]). Let be a transcendental entire function and . Then, there exists a set that has infinite logarithmic measure, such thatwhere is the central index of .

Lemma 3 (see [4]). Let be a rational function with -th order pole at . If is a meromorphic solution of the differential equationthen .

Lemma 4 (see [4]). Let be a meromorphic function, and let be some rational functions. If is a meromorphic solution of the equationsatisfying , then .

Lemma 5 (see [6]). Let be a meromorphic function with , and let be a given constant. Then, there exists a set that has finite logarithmic measure, such thatholds for all satisfying .

Lemma 6. Let be some meromorphic functions and . For , there exists a set that has finite logarithmic measure, such that for all satisfying , we havewhere are positive constants. If there is an entire function satisfying the equationthen .

Proof. Suppose that , is the maximum term of when , and is the central index of .
Using a similar method as in the proof of Lemma 2 in [5], we know that there exists a subset with infinite logarithmic measure, such thatSince is a step function of , without loss of generality, we suppose that are discontinuity points of and has a fixed central index when . Hence, for , we haveoutside of finite points in . Since is a continuous function, for , we haveBy Cauchy’s inequality, we have . From the inequality above, we can get thatChoosing a sufficiently large such thatBy (16)–(19), we can see that for sufficiently large , , we havewhere are the positive constants.
From Wiman–Valiron theory (see [7]), we can choose , such that we havewhere is of finite logarithmic measure.
By Lemma 1, we can see that there exists a set of finite logarithmic measure, such that for all satisfying , we haveNow, we can choose sufficiently large , is of infinite logarithmic measure; by (15) and (23), we havethat is,By , (20)–(22), and (24), we havewhere . For , we haveholds for . Hence, On the other hand, by (20) and (21), we can see that for sufficiently large , , we haveBy (29) and (30), we can know that (26) implies a contradiction. Hence, .

Lemma 7. Let be a rational function with -th order pole at , and let be a meromorphic function with . If is a meromorphic solution of (4), then(a)If , then all meromorphic solutions of (4) satisfy , with at most one exceptional solution satisfying .(b)If , then .

Proof. To our aim, we will consider the following three cases. Case 1. Suppose that , where is a rational function and is a polynomial and then . By Lemma 3 in [4], we can know that the conclusion holds. Case 2. Suppose that is a meromorphic function with infinitely many poles. Then, by (4), we have .(a)First, we will prove that if , then . Suppose that is a pole of of order and are all analytic at . Then, must be a pole of of order , which contradicts the fact that is analytic at . Hence, all poles of come from poles of and . Since has only finitely many poles, we can see that all poles of come from poles of with finitely many exceptions. Suppose that is a pole of of order and is analytic at , then is a pole of of order . Thus, we can know that has infinitely many poles. From we can see that and have the same exponent of convergence of poles, that is, By Hadamard factorization theory, we can set , where is a nonnegative integer, is an entire function, and , where is the canonical product formed with the nonzero poles of . . By , we have . By Lemma 1, we can choose , such that holds, and there exists a set of finite logarithmic measure, such that for all satisfying , we have By (32), we can know that and have the same exponent of convergence of poles. From the above proofs, we can see that except for finitely many exceptions, and have the same poles with the difference of their orders being ; hence, the above set concerning , for , we still have for all satisfying , holds. Substituting into (4), we get where are some differential polynomials, with constant coefficients, in . By and Lemma 5, we can see that there exists a set with finite logarithmic measure, such that for all satisfying , we have By (33), we have . Hence, by (37), we can easily obtain where are two nonzero constants and . Thus, by Lemma 6, (36) and (38), we have . By Lemma 2 and , we can see that there exists a set with infinite logarithmic measure, such that where is the central index of . Choosing , from Wiman–Valiron theory, we can know that (23) holds outside a set with finite logarithmic measure. For sufficiently large , by (39), we have where . Since is entire, , combining (35), (40), and , we have where . Choosing such that , since is of infinite logarithmic measure, by (23), (36), (38), and (41), we can see that for , we have By (39), we can see that for , we have By (43), we can see that the degree of terms of the left-hand side of (42) in is By (33), we have , which implies that Thus, comparing the degree of terms of both sides of (42) in , we obtain , which implies , that is, . Now, we will prove that all meromorphic solutions of (4) satisfy , with at most one exceptional solution satisfying . Suppose that and are two meromorphic solutions of (4) and satisfy . Then, . Since is a meromorphic solution of homogeneous equation (11) corresponding to (4), by Lemma 3, we have , which is a contradiction. Hence, equation (4) has at most one exceptional solution satisfying , and all other meromorphic solutions are satisfying .(b)Since , we will just prove that is false. Suppose that , and set , where have the same meanings as in (a). Then, using a similar method as in the proof of (a), we can see that for any given , (34)–(37) hold. By and (37), we can see that for , we have where are the two nonzero constants. By using (36) and (46) in conjunction with Lemma 6, we have . Continuing using a similar method as in the proof of (a), we can see (39)–(43) hold. Hence, for , we can see that in the left-hand side of (42), only one term has the highest degree in . It is impossible. Hence, . Case 3. Suppose that is a meromorphic function with finite many poles and infinite many zeros. Then, we can use a similar method as in the proof of Case 2 to obtain our conclusions.Using a similar method as in the proof of (a) in Lemma 7, we can obtain the following lemma.

Lemma 8. Let be a positive integer, and let be some rational functions with -th order pole at , where . is a meromorphic function, and . If the differential equationhas meromorphic solution, then all meromorphic solutions of (47) satisfy , with at most one exceptional solution satisfying .

Lemma 9. Let be rational functions and has a pole of order at , and is a polynomial with . If is a meromorphic solution of the equationthen .

Proof. Without loss of generality, we suppose that is transcendental. Otherwise, is a rational function, and we have .
Differentiating both sides of (48), we havethat is,where .
Since the poles of just appear at the poles of , has finitely many poles. Using to denote the sum of the major part of all poles of , is a transcendental entire function and . Substituting into (50), we haveSince the right-hand side of (51) is a rational function, its order is finite.
Suppose that and , where are some nonzero constants. Then, hence, there must exist some positive constants such thatApplying Lemma 6 on (51) and (53), we have . Hence, .

Lemma 10. Let be some rational functions with -th order pole at , and let be a meromorphic function with . If the equationhas meromorphic solution , then

Proof. Since have only finitely many poles, we can see that if is not a pole of , then is an -th order pole of if and only if is an -th order pole of ; hence, . Sincewe haveThus,Now, we will prove that every meromorphic solution of (54) satisfies . So, we will consider the following three cases: Case 1. Suppose that , where is a rational function and is a polynomial. Then, . Thus, by Lemma 9, we have . Case 2. Suppose that is a meromorphic function with infinitely many poles. Then, by (54), we have . If , then . Without loss of generality, we suppose that . Set , where have the same meanings as in the proof of (a) of Lemma 7. Using a similar method as in the proof of (a) of Lemma 7, we can get and . Substituting into (54), we have where are some differential polynomials, with constant coefficients, in , and . Using a similar method as in the proof of (a) of Lemma 7, we know (37) holds obviously. Hence, we can easily obtain where is a nonzero constant, satisfies , and . By Lemma 6 and (59) and (60), we have . Hence, . Case 3. Suppose that is a meromorphic function with finitely many poles and infinitely many zeros. Then, using a similar method as in the proof of Case 2, we can get .Thus, we can write in the following form:where are the integers, are the canonical product formed with the nonzero zeros of and , are the canonical product formed with the nonzero poles of and , and are the polynomials such that and . Substituting (61) into (54), we havewhere is a rational function in , , and its derivative, with constant coefficients. Comparing the growth order of both sides of (62) and noting that are canonical products, we havethat is,

3. Proofs of Theorems

The proof of Theorem 3. (a)By Lemma 7, we can see that all meromorphic solutions of equation (4) are satisfying , with at most one exceptional meromorphic solution satisfying . From Lemma 4, we can know that meromorphic solutions of (4) with satisfy .(b)By Lemmas 7 and 10, we get the required result.

The proof of Theorem 4. (a)By Lemma 7, we have .(b)If , then, by Lemma 10, we have .(c)If , set , where , is an integer and and are the canonical product (or polynomial) formed with the nonzero zeros and nonzero poles of , respectively. , and is a polynomial with . Setting , where is a meromorphic function. Thus, and have the same zeros and poles. Substituting into (4), we havewherewhere are the binomial coefficients and are the polynomials in and its derivatives, with constant coefficients and having degree . It is easy to see the derivative of with respect to having the same form with . Since , we know . By and Lemma 8, we can see that all meromorphic solutions of (65) satisfy , with at most one exceptional meromorphic solution satisfying . By Lemma 4, we have . From Lemma 10, we can know that satisfiesHence, (4) has at most one exceptional meromorphic solution satisfyingand all other meromorphic solutions satisfy .

4. Conclusion

Our paper investigates the nonhomogeneous linear differential equation , where is a rational function, having a pole at of order , and is a meromorphic function with finite order, and obtains some properties related to the order and zeros of its meromorphic solutions.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have drafted the manuscript and read and approved the final manuscript.

Acknowledgments

This research was supported by the Guangdong Young Innovative Talents Project (No. 2018KQNCX117) and by the National Natural Science Foundation of China (Nos. 11701524, 11771090, 11761035, and 11871260).