Research Article | Open Access
Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations
This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomials generated by solutions of the above equation, where and are entire functions that are not all equal to zero.
1. Introduction and Main Results
A function is called meromorphic if it is nonconstant and analytic in the complex plane except at possible isolated poles. If no poles occur, then reduces to an entire function. Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna’s value distribution theory of meromorphic functions, for reference see . In addition, we use notations and to denote the order and the exponent of convergence of zero sequence and to denote the sequence of distinct zeros of , respectively. A meromorphic function is called a small function with respect to if as , possibly outside of a set of with finite measure, where is the Nevanlinna characteristic function of .
For the second-order linear differential equation where is an entire function of finite order, it is well known that each solution of (1.1) is an entire function, and that if and are any two linearly independent solutions of (1.1), then at least one of must have infinite order, see [2, pages 167-168].
Thus, a natural question is the following: what condition on will guarantee that every solution of (1.1) has infinite order? Many researchers have studied the question, for the details see [3, page 291]. For the case that is a transcendental entire function, Gundersen  proved that if , then every solution of (1.1) has infinite order. In 2002, Chen considered the problem and proved the following result which is an improvement of Gundersen’s result.
Theorem A (see ). Let , be nonzero complex numbers satisfying and or , and let be entire functions with , then every solution of the equation
has infinite order.
Some further results on (1.2) were obtained for several cases. Chen  got the same conclusion when , and Chen and Shon  investigated the more general equations with meromorphic coefficients. Under the same assumption of Theorem A, if and are meromorphic functions with , then there is the same conclusion with Theorem A. In 2008, Wang and Laine  extended Theorem A to nonhomogeneous second-order linear differential equations.
Theorem B (see ). Let and be entire functions with , and let be complex constants that satisfy and , then every solution of differential equation is of infinite order.
Remark 1.1. Belaïdi and El Farissi  also proved Theorem B and got . We note that (2.21) in  cannot be deduced by following their proof. Indeed, as , holds just for the points satisfying , not for all . However, the difficulty can be got over by using Lemmas 2.5 and 2.6 in , and the method can be used in our proof of the following Theorem 1.2.
Since the beginning of the last four decades, a substantial number of research papers have been written to describe the fixed points of general transcendental functions. However, there are few studies on the fixed points of solutions of the general differential equations. In 2000, Chen  first studied the problems on the fixed points of solutions of second-order linear differential equations with entire coefficients. Since then, many results on fixed points of solutions of differential equations with entire coefficients were obtained, see [10–12]. In 2006, Chen and Shon  further studied the relation between small functions and solutions or differential polynomials of solutions of differential equations and obtained the following.
Theorem C. Let be entire functions with , and let be complex constants such that and or . If is an entire function with finite order, then every solution of (1.2) satisfies . Furthermore, let ,and be polynomials that are not all equal to zero, and let . If the order of is less than 1, then .
Belaïdi and El Farissi  also studied the relation between small functions and some differential polynomials generated by solutions of the second-order nonhomogeneous linear differential equation (1.3). They obtained the following.
Theorem D. Let and be entire functions with , and let be complex constants that satisfy and or . Let be entire functions that are not all equal to zero with , and let be an entire function with finite order. If is a solution of (1.3), then the differential polynomial satisfies .
The main purpose of this paper is to study the growth and the oscillation of solutions of second-order linear differential equation with meromorphic coefficients. Also, we will investigate the relation between small functions and differential polynomials generated by solutions of the above equation. Our results can be stated as follows.
Theorem 1.2. Let () and be meromorphic functions with , and let , be polynomials with degree (), where , (), are complex constants such that or , then every meromorphic solution of the equation has infinite order and satisfies
Theorem 1.3. Under the assumption of Theorem 1.2, and let be meromorphic functions that are not all equal to zero with , and let be a meromorphic function with finite order, if is a meromorphic solution of (1.4), then the differential polynomial satisfies .
Remark 1.4. Clearly, the method used in linear differential equations with entire coefficients cannot deal with the case of meromorphic coefficients. In addition, the proof of the results in [7, 13] relies heavily on the idea of Lemma 5 in  or Lemma 2.5 in . However, it seems too complicated to deal with our cases. We will use an important result in uniqueness theory of meromorphic functions, that is Lemma 2.5, to prove our theorems.
2. Preliminary Lemmas
In order to prove our theorems, we need the following lemmas.
Lemma 2.1 (see ). Let be a transcendental meromorphic function with . Let be a finite set of distinct pairs of integers satisfying for . Also let be a given constant, then there exists a set that has finite logarithmic measure, such that for all satisfying and for all , one has
Lemma 2.2 (see ). Let be a nonconstant polynomial of degree . Let be a meromorphic function, not identically zero, of order less than , and set . Then for any given there exists a zero measure set such that if , then for ,(1)if , then ,(2)if , then , where is a finite set.
Lemma 2.3 (see [8, Lemma 2.5]). Let be an entire function, and suppose that is unbounded on some ray with constant , then there exists an infinite sequence of points , where , such that and as .
Lemma 2.4 (see ). Let be finite-order meromorphic functions. If is an infinite-order meromorphic solution of the equation then satisfies .
Lemma 2.5 (see [17, page 79]). Suppose that are meromorphic functions and are entire functions satisfying the following conditions:(1),(2) are not constants for ,(3)for , , where has a finite measure.then .
Lemma 2.6 (see [8, Lemma 2.6]). Let be a an entire function of order . Suppose that there exists a set which has linear measure zero, such that for any ray , where is a positive constant depending on , while is a positive constant independent of . Then .
Proof. Suppose the contrary that , we will deduce a contradiction.
First, if , then . Clearly, , this is a contradiction.
Now suppose that . If , then By Lemma 2.5, we have , and this is a contradiction. Hence, .
Since is a meromorphic solution of (1.4), we know that the poles of can occur only at the poles of and . Let , where is the canonical product formed with the nonzero poles of , with , and is an entire function with . Substituting into (2.5), by some calculation we can get Now, we rewrite (2.6) into
Set . By Lemma 2.1, for any given , there exists a set which has linear measure zero, such that if , then there is a constant such that for all satisfying and , we have
Case 1. Suppose that , then by Lemmas 2.1 and 2.2, there exists a ray , and being defined in Lemma 2.2, such that , and for the above and sufficiently ,
Also, by Lemmas 2.1 and 2.2, we have
where is a constant.
Now we claim that is bounded on the ray . Otherwise, by Lemma 2.3, there exists a sequence of points , such that From (2.12) and the definition of order, we see that for is large enough. By (2.7), (2.8), (2.9), (2.10), and (2.13), we get where is a constant. Clearly, we can choose such that . Then by (2.14), we can obtain a contradiction. Therefore, is bounded, and we have on the ray .
Case 2. Suppose that . By Lemma 2.2, there exists a ray , where , , and are defined, respectively, as in Case 1, such that
Then, for any given , by Lemma 2.2 and (2.7), we have, for sufficiently large , As in Case 1, we prove that is bounded on the ray . Otherwise, similarly as in Case 1, there exists a sequence of points , such that , Further, we have for is large enough.
By (2.7), (2.8), (2.17), (2.18), and (2.21), we get Since and , we obtain a contradiction. So is bounded, and we have on the ray .
Combining Cases 1 and 2, for any given ray , of linear measure zero, we have (2.24) on the ray , provided that is sufficiently large. Thus by Lemma 2.6, we get , which is a contradiction. Then .
Proof. Suppose that is a meromorphic solution of (1.4), then by Theorem 1.2, we have .
Now suppose that . Substituting into , we have
Differentiating both sides of (2.25), and replacing with , we obtain Set Then we rewrite (2.25) and (2.26) into Set where are meromorphic functions formed by , and and their derivatives, with order less than , and is a index set. Since any one of is not equal to , then by Lemma 2.5, we have . This is a contradiction. Thus, .
By (2.28), we get If , then by (2.30) we have . Clearly, it is a contradiction. Hence, .
Suppose that or , and , then by similar discussion as above, we can get the same conclusion.
3. Proof of Theorem 1.2
4. Proof of Theorem 1.3
Suppose that or and that is a meromorphic solution of (1.4). Set . By and Lemma 2.8, we have . Without loss of generality, we assume that . Indeed, the remaining cases can be obtained by similar discussion. Substituting into (2.30), we have where is a meromorphic function of finite order. Then substituting (4.1) into (1.4), we have where are meromorphic functions formed by , and their derivatives. If , then by Theorem 1.2, we have . This is impossible, and hence . Thus, by , , (4.3), and Lemma 2.4, we get .
The author would like to thank the referees for their valuable suggestions and comments. This research was supported by NNSF of China (no. 11001057), NSF of Jiangsu Province (BK2010234), Project of Qinglan of Jiangsu Province.
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