Abstract

By means of the normal family theory, we estimate the growth order of meromorphic solutions of some algebraic differential equations and improve the related results of Barsegian et al. (2002). We also give some examples to show that our results occur in some special cases.

1. Introduction and Main Results

Let be a function meromorphic or holomorphic in the complex plane. We use the standard notations of Nevanlinna theory and denote the order of by (see Hayman [1], He and Xiao [2], and Laine [3] and Yang [4]).

Let be a domain in the complex plane. A family of meromorphic functions in is normal, if each sequence contains a subsequence which converges locally uniformly by spherical distance to a function meromorphic in ( is permitted to be identically infinite).

We define spherical derivative of the meromorphic function as follows:

An algebraic differential equation for is of the form where is a polynomial in each of its variables.

It is one of the important and interesting subjects to research the growth of meromorphic solution of differential equation (2) in the complex plane.

In 1956, Gol’dberg [5] proved that the meromorphic solutions have finite growth order when . Some alternative proofs of this result have been given by Bank and Kaufman [6] and by Barsegian [7].

In 1998, Barsegian [8, 9] introduced an essentially new type of weight for differential monomial below and gave the estimate for thefirst time for the growth order of meromorphic solutions of large classes of complex differential equations of higher degrees by using his initial method [10]. Later Bergweiler [11] reproved Barsegian’s result by using Zalcman’s lemma.

In order to state the result, we first introduce some notations [8]: , for , and put . Define by with the convention that . We call the weight of . A differential polynomial is an expression of the form where the are rational in two variables and is a finite index set. The total weight of is given by .

Definition 1. denotes the degree at infinity in variable concerning . : When all , we set .

In 2002, the following general estimate of growth order of meromorphic solutions of the equation was obtained, which depends on the degrees at infinity of coefficients of differential polynomial in , by Barsegian et al. [9].

Theorem A (see [9]). Let be a meromorphic solution to the differential equation , where . If or and , then the growth order of satisfies .

Remark 2. Barsegian [8, 12], Bergweiler [11], Frank and Wang [13], and Yuan et al. [14, 15] proved or the conditions hold for all .

In this paper, by the normal family method of Bergweiler in [11], we extend Theorem A and obtain the following result.

Theorem 3. Let , and let be a differential polynomial. If any meromorphic function whose all zeros have multiplicities at least satisfies the differential equations and or and , then the growth order of satisfies where is a polynomial of degree .

The following examples are to show that Theorem 3 is an extending result of Theorem A.

Example 4 (see [2]). For , let ; then and satisfies the following algebraic differential equation: When or , , and the growth order of any entire solution of (7) satisfies . When , , and the growth order of any entire solution of the above equation satisfies .

Example 5. For , the entire function satisfies the following algebraic differential equation: We know that , , , , , , and then .

Example 6. The entire function satisfies the following algebraic differential equation: We know that , , , , and the growth order of any entire solution of (9) satisfies .

Obviously, Example 6 shows that the result in Theorem 3 occurs.

Now we consider the similar result to Theorem 3 for the system of the algebraic differential equations: where and are two rational functions.

Qi et al. [16] obtained the following result.

Theorem B. Let , and let be a pair of meromorphic solutions of system (10). If , and all zeros of have multiplicity at least , then the growth orders of for satisfy

Remark 7. In 2009, Gu et al. [17] obtained Theorem B where , , , , and is a polynomial.

We obtain the following result.

Theorem 8. Let , and let be a pair of meromorphic solutions of system (10). If or and , and all zeros of have multiplicity at least , then the growth orders of for satisfy where and , if all , .

Example 9. The entire functions , satisfy an algebraic differential equation system: where is a constant, , , , , , , , and . So . It shows that the conclusion of Theorem 8 may occur.

Example 10. The entire functions , satisfy the following algebraic differential equation system: We know that , , , , , , , and the growth order of any meromorphic solution of (15) satisfies . It shows that Theorem 8 is an improvement result of Theorem B.

2. Main Lemmas

In order to prove our result, we need the following lemmas. Lemma 1 is an extending result of Zalcman [18] concerning normal family.

Lemma 1 (see [19]). Let be a family of meromorphic (or analytic) functions on the unit disc. Then is not normal on the unit disc if and only if there exist(a)a number ;(b)points with ;(c)functions ;(d)positive numbers ,such that converges locally uniformly to a nonconstant meromorphic (or entire) function , and its order is at most 2. In particular, we may choose and , such that

Lemma 2 (see [14]). Let be meromorphic in whole complex plane with growth order ; then for each , there exists a sequence , such that

3. Proofs of Theorems

Proof of Theorem 3. Suppose that the conclusion of theorem is not true; then there exists a meromorphic entire solution that satisfies the equation , such that By Lemma 2 we know that for each , there exists a sequence of points , such that (17) is right. This implies that the family is not normal at . By Lemma 1, there exist sequences and such that and converges locally uniformly to a nonconstant meromorphic function , whose order is at most 2, and all zeros of have multiplicity at least . In particular, we may choose and , such that According to (17) and (18)–(20), we can get the following conclusion.
For any fixed constant , we have
In the differential equation , we now replace by . Assuming that has the form (4), then we obtain
From we have where is a polynomial in two variables, whose degree in satisfies .
Hence we deduce that
Therefore, for every fixed , , and is not the zero and pole of , both (18) and < imply that ≤ = ≤ < , and then by (21), converges local uniformly as . Both and give that by (19), converges local uniformly as . Both (26) and (27) deduce that from (25) as . Since all zeros of have multiplicity at least , this is a contradiction.
The proof of Theorem 3 is complete.

Proof of Theorem 8. By the first equation of the systems of algebraic differential equations (7), we know Therefore we have
If is a rational function, then must be a rational function, so that the conclusion of Theorem 8 is right. If is a transcendental meromorphic function, by the system of algebraic differential equations (7), then we have
By applying Theorem 3 to (30), we know that the conclusions of Theorem 8 hold.
The proof of Theorem 8 is complete.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Authors’ Contribution

Wenjun Yuan and Jianming Lin carried out the design of the study and performed the analysis. Weiling Xiong participated in its design and coordination. All authors read and approved the final paper.

Acknowledgments

This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University. The first author would like to express his hearty thanks to Chern Institute of Mathematics for providing him with very comfortable research environments as he worked as a visiting scholar. The authors finally wish to thank the referee and the managing editor for their helpful comments and suggestions. This work was supported by the NSF of China (11271090) and the NSF of Guangdong Province (S2012010010121).