- Research Article
- Open Access
On the Growth of Solutions of Some Second-Order Linear Differential Equations
© Feng Peng and Zong-Xuan Chen. 2011
- Received: 10 December 2010
- Accepted: 9 February 2011
- Published: 6 March 2011
- Complex Number
- Entire Function
- Meromorphic Function
- Natural Question
- Standard Notation
Thus, a natural question is what conditions on and will guarantee that every solution of (1.2) has infinite order? Ozawa , Gundersen , Amemiya and Ozawa , and Langley  have studied the problem with and is complex number or polynomial. For the case that , and is transcendental entire function, Gundersen proved the following in [5, Theorem A].
has infinite order.
Theorem A states that when , (1.3) may have finite-order solutions. We go deep into the problem: what condition in when will guarantee every solution of (1.3) has infinite order? And more precise estimation for its rate of growth is a very important aspect. Chen investigated the problem and obtain the following in [8, Theorem B and Theorem C].
has infinite order.
For Theorems B and C, many authors, Wang and Lü ,Huang, Chen, and Li , and Cheng and Kang  have made some improvement. In this paper, we areconcerned with the more general problem, and obtain the following theorem that extend and improve the previous results.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Suppose that are real numbers, is a polynomial with degree , that is an entire function with . Set . Then for any given , there exists a set that has the linear measure zero, such that for any , there is , such that for , we have
Using Lemma 2.2, we can prove Lemma 2.3.
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Lemma 2.8 (see ).
Using the same reasoning as in Case 1(ii), we can get a contradiction.
Theorem 1.1 is thus proved.
This project was supported by the National Natural Science Foundation of China (no. 10871076).
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