- Open Access
On the growth of solutions of a class of second-order complex differential equations
© Yi et al.; licensee Springer 2013
- Received: 19 March 2013
- Accepted: 29 May 2013
- Published: 27 June 2013
In this paper, we consider the differential equation , where and are meromorphic functions, is a non-constant polynomial. Assume that has an infinite deficient value and finitely many Borel directions. We give some conditions on which guarantee that every solution of the equation has infinite order.
- complex differential equation
- meromorphic function
- Borel direction
- deficient value
where and are meromorphic functions, is a non-constant polynomial. We assume that the reader is familiar with the Nevanlinna theory of meromorphic functions and the basic notions such as , , and . For the details, see  or .
It is well known that if and are transcendental entire functions in equation (1) and , are two linearly independent solutions of equation (1), then at least one of , must have infinite order. Hence, ‘most’ solutions of equation (1) will have infinite order. On the other hand, there are some equations of the form (1) that possess a solution which has finite order; for example, satisfies the equation . Thus the main problem is what condition on and can guarantee that every solution of equation (1) has infinite order? There has been much work on this subject. For example, it follows from the work by Gunderson , Hellerstein et al.  that if and are entire functions with or is a polynomial and is transcendental; or if , then every solution of equation (1) has infinite order. Furthermore, if A is an entire function with finite order having a finite deficient value and is a transcendental entire function with , then every solution of equation (1) has infinite order . More results can be found in [6–9].
To state our theorem, we give some remarks first. Let () be a non-constant polynomial. Denote , let degP be the degree of , . In the following, we give the definition of the Borel direction of a meromorphic function .
Definition 1.1 
holds for any real number and every complex number with at most two exceptions.
The main results in this article are stated as follows.
Theorem 1.1 Let be a non-constant polynomial with , let be a meromorphic function with . Suppose that is a finite-order meromorphic function having an infinite deficient value, has only finitely many Borel directions: (). Denote that , . Suppose that there exists () such that for each angular domain . Then every meromorphic solution of equation (1) has infinite order and .
has infinite order with . Except the case of , by the theorem, we can get the part of the results above. But this theorem includes more general forms.
From the structure of and in , we can easily get the following conclusion.
then every meromorphic solution of equation (1) has infinite order and .
has infinite order with .
To prove our theorem, we need the following lemmas.
Lemma 2.1 
- (i)There exists a set that has linear measure zero, and there exists a constant that depends only on α and Γ such that if , then there is a constant such that for all z satisfying and , and for all , we have(4)
There exists a set that has finite logarithmic measure, and there exists a constant that depends only on α and Γ such that for all z satisfying and for all , the inequality (4) holds.
- (iii)There exists a set that has finite linear measure, and there exists a constant that depends only on α and Γ such that for all z satisfying and for all , we have(6)
Lemma 2.2 
- (i)If , then there is a constant such that the inequality(8)
- (ii)If , then there is a constant such that the inequality(9)
holds for .
Lemma 2.3 
is a non-negative and continuous function for with .
is a differentiable function for all r in with at most countable exceptions and .
The inequality holds for all sufficiently large r, and there exists a sequence with satisfying .
We shall call the function the proximate order of , and the function the type function of .
Lemma 2.4 
- (i)lies in the area of , , whose end points respectively for and (), and we have the following inequality:(11)
- (ii)For any positive number , the inequality(12)
holds for sufficiently large n and .
holds for all .
Since the whole complex plane is divided into q angular domains and there is no Borel direction in them, the circle is also divided into q arcs: ().
holds for all sufficiently large n.
The proof of Lemma 2.5 is completed. □
holds for all with and .
holds for all satisfying and .
Denote that , . Applying Lemma 2.5 to , then for any given constants and , there exists an angular domain and a sequence with () such that (13) holds for all sufficiently large m.
Obviously, when m is sufficiently large, this is a contradiction.
Next, we will prove .
As η can be arbitrary small, we have .
The proof of the theorem is completed. □
The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11171170, 61202313), the Natural Science Foundation of Jiang-Xi Province in China (Grant No. 2010GQS0119, No. 20132BAB211001 and No. 20122BAB201016).
- Hayman WK: Meromorphic Functions. Clarendon Press, Oxford; 1964.Google Scholar
- Yang L: Value Distribution Theory. Springer, Berlin; 1993.Google Scholar
- Gundersen GG: Finite order solution of second order linear differential equations. Trans. Am. Math. Soc. 1988, 305: 415-429. 10.1090/S0002-9947-1988-0920167-5MathSciNetView ArticleGoogle Scholar
- Hellenstein S, Miles J, Rossi J:On the growth of solutions of . Trans. Am. Math. Soc. 1991, 324: 693-705.Google Scholar
- Wu PC, Zhu J:On the growth of solution of the complex differential equation . Sci. China Ser. A 2011, 54(5):939-947. 10.1007/s11425-010-4153-xMathSciNetView ArticleGoogle Scholar
- Kwon KH: Nonexistence of finite order solution of certain second order linear differential equations. Kodai Math. J. 1996, 19: 378-387. 10.2996/kmj/1138043654MathSciNetView ArticleGoogle Scholar
- Chen ZX:The growth of where the order . Sci. China Ser. A 2002, 45(3):290-300.MathSciNetGoogle Scholar
- Chen ZX: On the hype-order of solution of some second order linear differential equations. Acta Math. Sin. Engl. Ser. 2002, 18(1):79-88. 10.1007/s101140100107MathSciNetView ArticleGoogle Scholar
- Laine I, Wu PC: Growth of solutions of second order linear differential equations. Proc. Am. Math. Soc. 2000, 128(9):2693-2703. 10.1090/S0002-9939-00-05350-8MathSciNetView ArticleGoogle Scholar
- Wu PC, Wu SJ, Zhu J: On the growth of solution of second order complex differential equation with moromorphic coefficients. J. Inequal. Appl. 2012., 2012(1): Article ID 117Google Scholar
- Zhang GH: The Theory of Entire Function and Meromorphic Function. Publication of China Science, Beijing; 1986.Google Scholar
- Gundersen GG: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 1988, 37: 88-104.MathSciNetGoogle Scholar
- Gao SA, Chen ZX, Chen TW: The Complex Oscillation Theory of Linear Differential Equation. Huazhong University of Science and Technology Press, Wuhan; 1997.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.