Abstract and Applied Analysis

Volume 2014 (2014), Article ID 123049, 4 pages

http://dx.doi.org/10.1155/2014/123049

Research Article

Algebroid Solutions of Second Order Complex Differential Equations

Lingyun Gao¹ and Yue Wang²

¹Department of Mathematics, Jinan University, Guangzhou, Guangdong 510632, China

²School of Information, Renmin University of China, Beijing 100872, China

Received 28 November 2013; Accepted 17 December 2013; Published 2 January 2014

Academic Editor: Zong-Xuan Chen

Copyright © 2014 Lingyun Gao and Yue Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Using value distribution theory and maximum modulus principle, the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that our results are sharp.

1. Introduction and Main Results

We use the standard notations and results of the Nevanlinna theory of meromorphic or algebroid functions; see, for example, [1, 2].

In this paper we suppose that second order algebraic differential equation (3) admit at least one nonconstant -valued algebroid solution in the complex plane. We denote by a subset of for which and by a positive constant, where denotes the linear measure of . or does not always mean the same one when they appear in the following.

Let    be entire functions without common zeroes such that . We put

Some authors had investigated the problem of the existence of algebroid solutions of complex differential equations, and they obtained many results ([2–10], etc.).

In 1989, Toda [4] considered the existence of algebroid solutions of algebraic differential equation of the form

He obtained the following.

Theorem A (see [4]). Let be a nonconstant -valued algebroid solution of the above differential equation and all are polynomials. If , then is algebraic.

The purpose of this paper is to investigate algebroid solutions of the following second order differential equation in the complex plane with the aid of the Nevanlinna theory and maximum modulus principle of meromorphic or algebroid functions:

where , .

We will prove the following two results.

Theorem 1. Let be a nonconstant -valued algebroid solution of differential equation (3) and all are polynomials. If , then is algebraic, .

Theorem 2. Let be a nonconstant -valued algebroid solution of differential equation (3) and the orders of all are finite. If , then the following statements are equivalent:

(a);(b);(c) is a Picard exceptional value of .

2. Some Lemmas

Lemma 3 (see [2]). Suppose that , , are meromorphic functions, and . Then one has

Examining proof of Lemma  4.5 presented in [2, pp. 192-193], we can verify Lemma 4.

Lemma 4. Let be a transcendental algebroid function such that has only finite number of poles, and let , , and have no poles in . Then, for some constants , , and it holds:

where .

Lemma 5 (see [11]). The absolute values of roots of equation

are bounded by

Lemma 6. Let be a nonconstant -valued algebroid solution of the differential equation (3) and let be a polynomial. If , then

where , is a positive constant.

Proof. We first prove that the poles of are contained in the zeroes of .

Suppose that is a pole of of order and is not the zeroes of . Then

We rewrite differential equation (3) as follows:

It follows from (10) that

Noting that , we have

This is a contradiction.

This shows that the poles of are contained in the zeroes of .

We rewrite differential equation (3) as follows:

For , we have

Applying Lemma 5 to (13) at ,

where .

From (14) and (15), we have

Note that

Dividing the inequality (17) by , we obtain, for ,

which reduces to our inequality by calculating of the both sides:

Lemma 6 is complete.

3. Proof of Theorem 1

First, we consider .

Let be a pole of of . Let be the order of zero of at .

(i) When the order of the pole of is not equal to that of other terms of the left-hand side of (10) at , we get

that is,

(ii) When the order of pole of is equal to that of some term of the left-hand side of (10) at , we get

that is,

Combining cases (i) and (ii), we obtain

where is a positive constant.

Secondly, by Lemma 6, we obtain

Combining the inequalities (25) and (26), we have

which shows that is an algebraic solution of (3).

This completes the proof of Theorem 1.

4. Proof of Theorem 2

(i) ( a) ⇒ (b). Suppose that . If , then we have by (3)

Applying Lemma 3 to (28),

Since is admissible solution, we have

so that

This is a contradiction. Thus, .

If , by Theorem 1, is nonadmissible. Thus,

(ii) ( b) ⇒ (c). Let . Then, similar to the proof of Lemma 6, we obtain that the poles of are contained in the set of and is a Picard exceptional value of .

(iii) ( c) ⇒ ( a). Let be a Picard exceptional value of . Then .

5. Some Examples

Example 1. The differential equation

has a transcendental algebroid solution . In this case

Remark 7. Example 1 shows that the condition in Theorem 1 is sharp.

Example 2. Transcendental algebroid function is a 2-valued solution of the following differential equation:

In this case

By Theorem 2, for transcendental algebroid function , is a Picard exceptional value.

Remark 8. Example 2 shows that the result in Theorem 2 holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project is project Supported by National Natural Science Foundation (10471065) of China and NSF of Guangdong Province (04010474).

References

H. Yi and C. C. Yang, Theory of the Uniqueness of Meromorphic Functions, Science Press, Beijing, China, 1995 (Chinese).

Y. Z. He and X. Z. Xiao, Algebroid Functions and Ordinary Differential Equations, Science Press, Beijing, China, 1988.

Y. Z. He and X. Z. Xiao, “Admissible solutions and ordinary differential equations,” Contemporary Mathematics, vol. 25, pp. 51–61, 1983.

N. Toda, “On algebroid solutions of some algebraic differential equations in the complex plane,” Japan Academy A, vol. 65, no. 4, pp. 94–97, 1989.

T. Chen, “One class of ordinary differential equations which possess algebroid solutions in the complex domain,” Chinese Quarterly Journal of Mathematics, vol. 6, no. 4, pp. 45–51, 1991.

K. Katajamäki, “Value distribution of certain differential polynomials of algebroid functions,” Archiv der Mathematik, vol. 67, no. 5, pp. 422–429, 1996.

L. Gao, “Some results on admissible algebroid solutions of complex differential equations,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 7, pp. 1041–1050, 2001.

L.-Y. Gao, “On some generalized higher-order algebraic differential equations with admissible algebroid solutions,” Indian Journal of Mathematics, vol. 43, no. 2, pp. 163–175, 2001.

L. Gao, “On the growth of solutions of higher-order algebraic differential equations,” Acta Mathematica Scientia B, vol. 22, no. 4, pp. 459–465, 2002.

L. Y. Gao, “The growth of single-valued meromorphic solutions and finite branch solutions,” Journal of Systems Science and Mathematical Sciences, vol. 24, no. 3, pp. 303–310, 2004.

T. Takagi, Lecture on Algebra, Kyoritsu, Tokyo, Japan, 1957 (Japanese).