Oscillation results on meromorphic solutions of second order differential equations in the complex plane

The main purpose of this paper is to consider the oscillation theory on meromorphic solutions of second order linear differential equations of the form f '' + A(z)f = 0 where A is meromorphic in the complex plane. We improve and extend some oscillation results due to Bank and Laine, Kinnunen, Liang and Liu, and others.


Introduction and main results
Let us define inductively, for r ∈ [0, +∞), exp 1 r = e r and exp n+1 r = exp(exp n r), n ∈ N.For all r sufficiently large, we define log 1 r = log + r = max{log x, 0} and log n+1 r = log(log n r), n ∈ N. We also denote exp 0 r = r = log 0 r, log −1 r = exp 1 r and exp −1 r = log 1 r.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 (e.g.see [8,21]), such as T (r, f ), m(r, f ), and N (r, f ).Throughout the paper, a meromorphic function f means meromorphic in the complex plane C. To express the rate of fast growth of meromorphic functions, we recall the following definitions (e.g.see [4,6,12,14,15,19]).Definition 1.2.The growth index (or the finiteness degree) of the iterated order of a meromorphic function f is defined by Definition 1.3.The iterated convergence exponent of the sequence of a-points of a meromorphic function f is defined by Definition 1.4.The growth index (or the finiteness degree) of the iterated convergence exponent of the sequence of a-points of a meromorphic function f with iterated order is defined by Remark 1.2.Similarly, we can use the notation λ n (f − a) to denote the iterated convergence exponent of the sequence of distinct a-points, and use the notation i λ (f − a) to denote the growth index of λ n (f − a).
It is well-known that Nevanlinna theory has appeared to be a powerful tool in the field of complex differential equations (see [1-7, 11-18, 20], for example) .The active research of the complex oscillation theory of linear differential equations in the complex plane C was started to investigate the second order differential equation ( 1) by Bank and Laine [1,2].They investigated this question in the case where A is an entire function, mainly from the point of determining the distribution of zeros of solutions.In this case all solutions of Eq.( 1) are entire.When A is meromorphic, there are some immediate difficulties.For example, it is possible that no solution of Eq.( 1) except the zero solution is single-valued on the plane.This obstacle was handled since Bank and Laine [2] gave necessary and sufficient conditions for all solutions of Eq.( 1) to be meromorphic, and hence single-valued, in a simplyconnected region.To consider poles as well as zeros, they obtained the following theorems.
In the special case where λ 1 ( 1 A ) < ∞ (e.g.A is of finite order), we can conclude that λ 1 (f ) = ∞ unless all solutions of Eq.( 1) are finite order.
Kinnunen obtained the following result, of this type corresponding to Theorem 3.1 in [3].
Thus it is interesting to consider the complex oscillation on the meromorphic solutions of the Eq.(1) for the case where A is meromorphic function in the terms of the idea of iterated order.In 2007, Liang and Liu [17] considered the complex oscillation on the Eq.(1) when A is a meromorphic function with finite many poles.By using the Wiman-Valiron theory (for an entire function [9,11], for a meromorphic function [7,20]), they obtained some results which extend Theorems 1.3 and 1.4.There arises naturally a question: Question 1.1.What can be said if A has infinitely many poles?
Although the Wiman-Valiron theory is a powerful tool to investigate entire solutions, it is only useful for the meromorphic function A with λ 1 ( 1 A ) < σ 1 (A) if considering the Eq.(1).In this paper we shall make use of a recent result due to Chiang and Hayman (see Lemma 2.3 in the next section) instead of the Wiman-Valiron theory, and thus answer the above question.In fact, we obtain the following results which improve and extend some oscillation results due to Bank & Laine [2], Liang & Liu [17] and others.Furthermore, considering the deficiencies of poles of the coefficient A and solutions f of Eq. ( 1), we obtain some special results.For a ∈ C = C ∪ {∞}, the deficiency of a with respect to a meromorphic function g in C is defined by provided that g has unbounded characteristic.The first result is the following Theorem 1.6.Let A be a meromorphic function with 0 < i(A) = n < ∞, and assume that λ n (A) < σ n (A) = 0.Then, if f is a nonzero meromorphic solution of the Eq. ( 1) we have In the special case where either δ(∞, f ) > 0 or the poles of f are of uniformly bounded multiplicities, we can conclude that Theorem 1.6 improves and extends Theorem 3.3 in [14] and Theorem 5 in [2].It is obvious from Theorem 1.6 that the following corollary is true, which improves and extends Corollary 3.4 in [14].
The next result improves and extends Theorem 6 and Corollary 7 in [2].
Theorem 1.7.Let A be a meromorphic function with 0 < i(A) = n < ∞.Assume that the Eq. ( 1) possesses two linearly independent meromorphic solutions f 1 and then any nonzero solution f of (1) which is not a constant multiple of either f 1 or f 2 satisfies, λ n (f ) = ∞, unless all solutions of (1) are of finite iterated n-order.In the special case where δ(∞, A) > 0, or , (e.g.A is an entire function), we can conclude that λ n (f ) = ∞.
We remark that Theorem 1.7 and the following theorem are the improvement and extension of Theorem 3.2 in [14].
Theorem 1.8.Let A be a meromorphic function satisfying 0 < i(A) = n < ∞.Assume that the Eq. ( 1) possesses two linearly independent meromorphic solutions , and if either δ(∞, f ) > 0 or the poles of f are of uniformly bounded multiplicities, then we have i λ (E) ≤ n + 1 and have From the proof of Theorem 1.8 one can get the following result.
Corollary 1.2.Let A be a meromorphic function satisfying Assume that the Eq. ( 1) possesses two linearly independent meromorphic solutions The following corollary is immediately obtained from Theorem 1.8 which is an improvement of Theorem 1.3.
Corollary 1.3.Let A be an entire function with 0 < i(A) = n < ∞.Let f 1 and f 2 be two linear independent solutions of Eq.( 1), and denote Finally, we show the following result which extends and improves Theorem 1.5.
Theorem 1.9.Let A be a meromorphic function with 1 < i(A) = n < ∞.Assume that f 1 and f 2 are two linearly independent meromorphic solutions of the Eq. ( 1) such that Let Π = 0 be any meromorphic function for which either i(Π) < n or σ n (Π) < σ n (A).Let g 1 and g 2 be two linearly independent solutions of the differential equation The remainder of this paper is organized as follows.Section 2 is for some lemmas and the other sections are for the proofs of our main results.The idea and formulations of our main results come from [1,2,14].The proof of Theorem 1.6 is from the proof of Theorem 5 in [2], the proof of Theorem 1.7 is essentially from the proof of Theorem 6 in [2], and the proof of Theorem 1.9 is a parallel to a corresponding reasoning in the proof of Theorem 3.6 in [14].

Some lemmas
To prove our results, we need the following lemmas.
where S(r, f ) := o(T (r, f )) as r → ∞ outside of a possible exceptional set of finite linear measure.
Lemma 2.3.([13], Theorem 6.2) Let f be a meromorphic solution of (5) where A 0 , . . ., A k−1 are meromorphic functions in the plane C. Assume that not all coefficients A j are constants.Given a real constant γ > 1, and denoting T (r) : outside of an exceptional set E p with Ep t p−1 dt < +∞.
We note that in the above lemma, p = 1 corresponds to Euclidean measure and p = 0 to logarithmic measure.Using logarithmic measure not Euclidean measure, we correct here the proof of Theorem 3.2 in [6].(5) we get that the poles of f (z) can only occur at the poles of A 0 , A 1 , . . ., A k−1 .Note that the multiplicities of poles of f are uniformly bounded, and thus we have where M 1 and M are some suitable positive constants.This gives (6) T (r, f ) = m(r, f ) + O(max{N (r, A j ) : j = 0, 1, . . ., k − 1}).
If δ(∞, f ) := δ 1 > 0, then for sufficiently large r, Applying now ( 6) or ( 7) with Lemma 2.3, we obtain outside of an exceptional set E 0 with finite logarithmic measure.Using a standard method to deal with the finite logarithmic measure set, one immediately gets from above inequalities that σ n+1 (f ) ≤ max{σ n (A j ) : j = 0, 1, . . ., k − 1}.Replacing the notation n(r, f ) by n(r, f ) and following the reasoning of the proof of Lemma 1.7 in [14], one can easily obtain the following lemma.
We recall here the essential part of the factorization theorem for meromorphic functions of finite iterated order.Lemma 2.7.( [12], Satz 12.4) A meromorphic function f for which i(f ) = n can be represented by the form where U, V and g are entire functions such that The following result plays a key role in the present paper, which is an improvement and extension of Theorem 3.1 in [14] and Theorem 1 in [17].
Lemma 2.8.Let A be a meromorphic function with i(A) = n (0 < n < ∞), and let f be a nonzero meromorphic solution of the Eq.(1).Then (i) if either δ(∞, f ) > 0 or the poles of f are of uniformly bounded multiplicities, then i(f Proof.Assume that f is a nonzero meromorphic solution of the Eq. ( 1).It is obvious that (i) is just a special case of Lemma 2.4.
Following Hayman [10], we shall 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 theorems, see the following sections.

Proof of Theorem 1.6
Since f is a solution of (1) where σ n (A) > 0, it is obvious that f can not be rational, nor be of the form e az+b for constants a and b.Hence, by Lemma 2.1 we have In addition, by (1) we have By assumption, λ n (A) < σ n (A).Hence, if we assume that (2) fails to hold, then we deduce by ( 15) and ( 16) that σ n f f ′ < σ n (A).By the first main theorem, we then . However, from (1) it easily follows that EJQTDE, 2010 No. 68, p. 8 , and so we obtain σ n (A) ≤ σ n (ϕ) < σ n (A), a contradiction.Hence, (2) is true.
In the special case where either δ(∞, f ) > 0 or the poles of f are of uniformly bounded multiplicities, by Lemma 2.8 we have Hence, we obtain (3).
From (1), we see that any pole of A is at most double and is either a zero or pole of f, we thus have Hence by assumption, N (r, A) = O exp n−1 (r d ) as r → ∞ for some d > 0. Together with ( 18) and ( 19), we obtain N (r, A) = O exp n−1 (r d ) n. e. as r → ∞, from which it follows by standard reasoning that f 1 is of finite iterated n-order.By the identity of Abel, we have where β is equal to the Wronskian of f 1 and f 2 .Hence, by Lemma 2.2 and (20), we obtain Reversing the roles of f 1 and f 2 , we can conclude that σ n (f 1 ) = σ n (f 2 ).Hence, all solutions of (1) are of finite iterated n-order if λ n (f ) < ∞.
By the assumption (4), we have for some β < σ n (A) and the iterated order of the function A implies that T (r, A) = O exp n−1 (r σn(A)+ε ) .