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On the growth of solutions of a class of second-order complex differential equations
Advances in Difference Equations volume 2013, Article number: 188 (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.
1 Introduction and main results
In this paper, we shall involve the deficient value and the Borel direction in investigating the growth of solutions of the second-order linear differential equation
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 .
The order σ and the hyper-order are defined as follows:
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].
However, it seems that there is little work done on equation (1) whose coefficient functions are meromorphic functions. Recently, Wu et al. discussed the problem correlating with this in . Now we still consider equation (1) with transcendental meromorphic coefficients and discuss the growth of its meromorphic solutions. We shall also involve the deficient value and the Borel direction in the studies of the oscillation of the second-order complex differential equation. We hope that the relations between the orders of coefficient functions will not be restricted. In general, it would not hold that every solution of equation (1) has infinite order; for example, satisfies
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 
Let be a meromorphic function in the complex plane with (). A ray () starting from the origin is called a Borel direction of order σ of if the following equality:
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 .
Remark 1.1 We apply the theorem to some particular equations. For example, when , where is a non-constant polynomial and . Chen proved  that every meromorphic solution of the equation
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.
Corollary 1.2 Let be a non-constant polynomial with , let be a meromorphic function with . Let be a transcendental meromorphic function with finite order. If has a deficient value ∞ and has only q Borel directions () that satisfy () and
then every meromorphic solution of equation (1) has infinite order and .
By using the corollary, we see that if , then every meromorphic solution of the equation
has infinite order with .
2 Some lemmas
To prove our theorem, we need the following lemmas.
Lemma 2.1 
Let denote a pair that consists of a transcendental meromorphic function and a finite set
of distinct pairs of integers that satisfy for . Let and be given real constants. Then the following three statements hold.
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)
In particular, if has finite order , then (4) is replaced by (5).
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.
In particular, if has finite order , then the inequality (5) holds.
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)
In particular, if has finite order , then (6) is replaced by (7)
Lemma 2.2 
Suppose that , where is a non-constant polynomial with , and is a meromorphic function with . There exists a set that has linear measure zero such that for all , we have
If , then there is a constant such that the inequality(8)
holds for .
If , then there is a constant such that the inequality(9)
holds for .
Lemma 2.3 
Let be a transcendental meromorphic function with finite order σ, then there exists a function with the following properties:
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 
Let be a transcendental meromorphic function with order σ (). and () are two half rays starting from the origin, and f has no Borel direction in the angular domain . Suppose that there exists a sequence with () and a complex number () such that the following inequality:
holds for any given constant and all sufficiently large n in some rays , where . We denote the arc and the angular set such that any satisfies the inequality (10). If there exists a constant (not dependent on ε) such that , then we can get a list of curve segment satisfying the following two conditions for any given () and sufficiently small :
lies in the area of , , whose end points respectively for and (), and we have the following inequality:(11)
For any positive number , the inequality(12)
holds for sufficiently large n and .
Lemma 2.5 Let be a transcendental meromorphic function with order σ having an infinite deficient value. If has q Borel directions, (), and these half-rays divide the whole complex plane into q angular domains, , , , then for any given constant and , there exists an angular domain at least and a sequence with () such that the following inequality:
holds for all sufficiently large n, where
Proof Let be a proximate order of with a type function . According to the properties of of Lemma 2.3, there exists a sequence with satisfying . Let ( ()) be all the poles of in . For every , by the Boutroux-Cartan theorem , we have
except for a set of points that can be enclosed in a finite number of disks with the sum of total radius not exceeding . Set . Then, for every integer n, we can choose satisfying . By the Poisson-Jensen formula and (14), for any z satisfying , we have
And then, we have
In addition, since , there exists a constant such that the inequality
holds for all .
According to the properties of , we have
From (15)-(17), we get
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: ().
Obviously, we have
where . Hence, by (16) and the properties of , for any given , there exist and a sequence with () (otherwise, we use the subsequence instead of ) such that the following inequality:
holds for all sufficiently large n.
We choose , . By Lemma 2.4, for all sufficiently large n, there exists a curve such that (11) and (12) hold. So, for any given , we have
The proof of Lemma 2.5 is completed. □
3 Proof of Theorem 1.1
By Lemma 2.1(i), there exists a set of measure zero and such that the following inequality:
holds for all with and .
Suppose that , where . We have by calculation. By Lemma 2.2, there exists a set of measure zero and such that the following inequality:
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.
On the other hand, since there exists in such that by the supposition of the theorem, we can get an interval such that (20) holds for all satisfying and . Now, let . For each sufficiently large m, we can choose such that (19), (20) and the inequality
hold for . Let . Hence, from (18)-(21), we get
Obviously, when m is sufficiently large, this is a contradiction.
Next, we will prove .
By using Lemma 2.1, there exist a set of measure zero and two constants and such that for all z satisfying and , the following inequality holds:
Hence, for each sufficiently large m, we can choose such that (20), (21) and (23) hold for . From (18), (20), (21) and (23), we get
As η can be arbitrary small, we have .
The proof of the theorem is completed. □
Hayman WK: Meromorphic Functions. Clarendon Press, Oxford; 1964.
Yang L: Value Distribution Theory. Springer, Berlin; 1993.
Gundersen GG: Finite order solution of second order linear differential equations. Trans. Am. Math. Soc. 1988, 305: 415-429. 10.1090/S0002-9947-1988-0920167-5
Hellenstein S, Miles J, Rossi J:On the growth of solutions of . Trans. Am. Math. Soc. 1991, 324: 693-705.
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-x
Kwon KH: Nonexistence of finite order solution of certain second order linear differential equations. Kodai Math. J. 1996, 19: 378-387. 10.2996/kmj/1138043654
Chen ZX:The growth of where the order . Sci. China Ser. A 2002, 45(3):290-300.
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/s101140100107
Laine I, Wu PC: Growth of solutions of second order linear differential equations. Proc. Am. Math. Soc. 2000, 128(9):2693-2703. 10.1090/S0002-9939-00-05350-8
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 117
Zhang GH: The Theory of Entire Function and Meromorphic Function. Publication of China Science, Beijing; 1986.
Gundersen GG: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 1988, 37: 88-104.
Gao SA, Chen ZX, Chen TW: The Complex Oscillation Theory of Linear Differential Equation. Huazhong University of Science and Technology Press, Wuhan; 1997.
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).
The authors declare that they have no competing interests.
CFY and XQL completed the main part of this article, CFY, XQL and HYX corrected the main theorems. All authors read and approved the final manuscript.
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Yi, C.F., Liu, X. & Xu, H.Y. On the growth of solutions of a class of second-order complex differential equations. Adv Differ Equ 2013, 188 (2013). https://doi.org/10.1186/1687-1847-2013-188
- complex differential equation
- meromorphic function
- Borel direction
- deficient value