The Oscillation on Solutions of Some Classes of Linear Differential Equations with Meromorphic Coefficients of Finite [p, q]-Order

This paper considers the oscillation on meromorphic solutions of the second-order linear differential equations with the form f′′ + A(z)f = 0, where A(z) is a meromorphic function with [p, q]-order. We obtain some theorems which are the improvement and generalization of the results given by Bank and Laine, Cao and Li, Kinnunen, and others.


Introduction and Main Results
The purpose of this paper is to study the oscillation on solutions of linear differential equations in the complex plane. It is well known that Nevanlinna theory has appeared to be a powerful tool in the field of complex differential equations. We assume that readers are familiar with the standard notations and the fundamental results of the Nevanlinna's value distribution theory of meromorphic functions (see [1][2][3][4]). Throughout the paper, a meromorphic function means meromorphic in the complex plane C. In addition, we use ( ) and ( ) to denote the order and the exponent of convergence of zero sequence of meromorphic function ( ), respectively. For sufficiently large ∈ [1,∞), we define log +1 = log (log ) ( ∈ ) and exp +1 = exp(exp ) ( ∈ ) and exp 0 = = log 0 , exp −1 = log .
For the second-order linear differential equation, where ( ) is an entire function or meromorphic function of finite order. In 1982, Bank and Laine [5] mainly studied the distribution of zeros of solutions of (1) when is an entire function of finite order. Obviously, all solutions of (1) are entire when ( ) is entire. However, there are some immediate difficulties when is meromorphic; for example, the solutions of (1) may not be entire, and it is possible that no solution of (1) except the zero solution is single-valued on the plane. In 1983, Bank and Laine [6] investigated the exponent of convergence of zero sequence of nontrivial solutions of (1), when is meromorphic function and obtained the results as follows.
After their work, many authors have investigated the growth and the exponent of convergence of zero sequence of non-trivial solutions of (1) and obtained many classical results (see [5,7,8]).
In 1998, Kinnunen [9] further investigated the oscillation results of entire solutions of (1) when ( ) is an entire function with finite iterated order and obtained some theorems which improved some theorems given by Bank and Laine [5]. Later, Chen [10] and Wang and Lü [11] studied the oscillation of solutions of (1) when ( ) is a meromorphic function with finite order by using the Wiman-Valiron theory; they obtained some results which extend some theorems of Kinnunen [9]. In 2007, Liang and Liu [12] considered the complex oscillation on (1) when ( ) is a meromorphic function with many finite poles. They extended the above oscillation results by using the method of the Wiman-Valiron theory. Although the Wiman-Valiron theory is a powerful tool to investigate entire solutions, it is only useful for the meromorphic function ( ) with the exponent of the convergence sequence of poles which is less than the order of ( ) if we consider (1). In 2010, Cao and Li [13] made use of a result due to Chiang and Hayman [14] instead of the Wiman-Valiron theory and obtained four oscillation theorems and three corollaries which extended the above results due to Bank-Laine, Kinnunen, and Liang and Liu.
Thus, it is interesting to consider the complex oscillation on the meromorphic solutions of (1) for the case when is entire or meromorphic functions in the terms of the idea of [ , ]-order.
In this paper, we further investigated the complex oscillation of meromorphic solutions of (1) when ( ) is meromorphic by using the idea of [ , ]-order. To state our theorems, we first introduce the concepts of entire functions of [ , ]order (see [15,16,18]). Throughout this paper, we always assume that , are positive integers satisfying ≥ ≥ 1.
(i) If 2 − 1 > 2 − 1 , then the growth of 1 is slower than the growth of 2 .
(iii) If 2 − 1 = 2 − 1 > 0, then the growth of 1 is slower than the growth of 2 if 2 ≥ 1 while the growth of 1 is faster than the growth of 2 if 2 < 1.
Definition 11 (see [15,16,18]). The [ , ] exponent of convergence of the zero sequence and the [ , ] exponent of convergence of the distinct zero sequence of ( ) are defined respectively, by log .

(7)
Remark 12. It is easy to know that Now, we will show our main results on the complex oscillation on meromorphic solutions of (1) when ( ) is meromorphic with finite [ , ]-order as follows.
Remark 17 (Following Hayman [20]). we will use the abbreviation "n.e. " (nearly everywhere) to mean "everywhere in (0, +∞) except in a set of finite measure" in the proofs of our main results of this paper.

Some Lemmas
For the proof of our results we need the following lemmas.
Using the same proof of Remark 1.3 in [9], one can easily prove the following lemma.
Thus we complete the proof of Lemma 22.

Lemma 24. A meromorphic function ( ) with [ , ] index can be represented by the form
outside of an exceptional set with ∫ −1 < +∞.
Remark 26. From the above lemma, we can see that = 1 corresponds to Euclidean measure and = 0 to logarithmic measure.
Using the above lemma, we can get the following lemma. Proof. Suppose that is a nonzero meromorphic solution of (1). It is easy to see that (i) holds since (i) is just a special case of Lemma 25. Next, we assume that satisfies (∞, ) > 0 or holds for any given (0 < < ).
We will consider two cases as follows.

Proof of Theorem 13
Since Suppose that (9) fails to hold that is Combining the above discussions, we can get (10). Thus, this completes the proof of Theorem 13.
The Scientific World Journal 7