- Research
- Open Access
On meromorphic solutions of some linear differential equations with entire coefficients being Fabry gap series
- Shun-Zhou Wu1 and
- Xiu-Min Zheng1Email author
https://doi.org/10.1186/s13662-015-0380-3
© Wu and Zheng; licensee Springer. 2015
- Received: 31 July 2014
- Accepted: 20 January 2015
- Published: 31 January 2015
Abstract
Keywords
- linear differential equation
- meromorphic solution
- growth
- exponent of convergence
- Fabry gap series
MSC
- 30D35
- 34M10
1 Introduction and main results
We make use of the standard notations of Nevanlinna’s value distribution theory (see, e.g., [1–3]). In the whole paper, let \(f(z)\) be a meromorphic function in the whole complex plane.
Firstly, let us recall the following definitions (see, e.g., [4–6]).
Definition 1.1
Definition 1.2
Definition 1.3
Further, we can get the definitions \(\lambda_{p}(f-\varphi)\) and \(\overline{\lambda}_{p}(f-\varphi)\), when a is replaced by a meromorphic function \(\varphi(z)\).
In 1993, Chen and Gao considered the growth of solutions of (1.1) and obtained the following theorem in [9].
Theorem A
(see [9])
- (i)
\(\sigma(A_{j})<\sigma(A_{0})<\infty\), \(j=1,\ldots,k-1\)
- (ii)
\(A_{0}(z)\) is a transcendental entire function with \(\sigma(A_{0})<\infty\), and \(A_{j}(z)\), \(j=1,\ldots,k-1\), are polynomials.
Generally, when \(A_{d}(z)\) (\(0\leq d\leq k-1\)) is dominant, Chen and Gao obtained the following theorem in [10] in 1997.
Theorem B
(see [10])
- (i)
\(\sigma(A_{j})<\sigma(A_{d})<\frac{1}{2} \) (\(j=0,\ldots,d-1,d+1,\ldots,k-1\))
- (ii)
\(A_{d}(z)\) is transcendental with \(\sigma(A_{d})=0\) and \(A_{j}(z)\) (\(j\neq d\)) are polynomials.
Theorems A and B give the properties of solutions of (1.1) when there is exactly one coefficient that has the maximal order. Thus, a natural question arises: how about the properties of solutions of (1.1) when there is more than one coefficient having the maximal order? In this paper, we proceed in this way.
Theorem 1.1
Remark 1.1
Suppose that \(A_{k}(z)=\sum_{n=0}^{\infty}c_{\lambda_{n}}z^{\lambda_{n}}\) is an entire function, and the sequence of exponents \(\{\lambda_{n}\}\) satisfies Fabry gap condition (1.3), then the series \(\sum_{n=0}^{\infty}c_{\lambda_{n}}z^{\lambda_{n}}\) is called a Fabry gap series. It follows by [16] that if \(A_{k}(z)\) is a Fabry gap series, then it has no deficient values. In particular, zero is not a deficient value of \(A_{k}(z)\), then the solutions of (1.2) are meromorphic in general.
Theorem 1.2
- (i)
\(\sigma(A_{j})<\sigma(A_{k})<\frac{1}{2}\), \(j=0,1,\ldots,k-1\)
- (ii)
\(A_{k}(z)\) is transcendental with \(\sigma(A_{k})=0\), and \(A_{j}(z)\), \(j=0,1,\ldots,k-1\), are polynomials.
Theorem 1.3
- (i)If \(\sigma(F)<\sigma(A_{k})\) (now \(A_{k}(z)\) does not satisfy (ii) of Theorem 1.2), then every rational solution \(f(z)\) of (1.4) is a polynomial with \(\deg f\leq k-1\), and every transcendental meromorphic solution \(f(z)\) of (1.4), whose poles are of uniformly bounded multiplicities such that \(\lambda(\frac{1}{f})<\mu(f)\), satisfieswhere \(\varphi(z)\) is a finite order meromorphic function and does not solve (1.4).$$\overline{\lambda}(f-\varphi)=\lambda(f-\varphi)=\sigma(f)=\infty,\qquad \overline{\lambda}_{2}(f-\varphi)=\lambda_{2}(f-\varphi)= \sigma_{2}(f)=\sigma(A_{k}), $$
- (ii)If \(\sigma(F)>\sigma(A_{k})\), then every infinite order meromorphic solution \(f(z)\) of (1.4) satisfieswhere \(\varphi(z)\) is a finite order meromorphic function and does not solve (1.4). And every finite order meromorphic solution \(f_{0}(z)\) satisfies$$\overline{\lambda}(f-\varphi)=\lambda(f-\varphi)=\sigma(f)=\infty, $$$$\sigma(F)\leq\sigma(f_{0})\leq\max\bigl\{ \sigma(F), \overline{ \lambda}(f_{0})\bigr\} . $$
For the case of entire solutions, we can deduce the following Corollary 1.1 easily.
2 Preliminary lemmas
Lemma 2.1
(see [17])
Lemma 2.2
(see [6])
Lemma 2.3
Proof
Lemma 2.4
We may deduce the following Remark 2.1 from Lemma 2.4 immediately.
Remark 2.1
Lemma 2.5
(see [6])
- (i)
\(\max\{i(F)=q, i(A_{j})\ (j=0,1,\ldots,k-1)\}< i(f)=p+1\) (\(0< p<\infty\)),
- (ii)
\(b=\max\{\sigma_{p+1}(F),\sigma_{p+1}(A_{j})\ (j=0,1,\ldots,k-1)\}<\sigma _{p+1}(f)=\sigma\).
Lemma 2.6
(see [12])
Suppose that \(k\geq2\), \(A_{j}(z)\), \(j=0,1,\ldots,k-1\), are meromorphic functions, \(\sigma=\max\{\sigma(A_{j}), j=0,1,\ldots,k-1\}\). If \(f(z)\) is a transcendental meromorphic solution of (1.1) and all poles of \(f(z)\) are of uniformly bounded multiplicity, then we have \(\sigma_{2}(f)\leq\sigma\).
Lemma 2.7
(see [17])
Lemma 2.8
(see [18])
Lemma 2.9
(see [19])
Lemma 2.10
(see [11])
Lemma 2.11
(see [20])
Let \(g: [0, +\infty)\rightarrow\mathbb{R}\) and \(h: [0, +\infty)\rightarrow\mathbb{R}\) be monotone nondecreasing functions such that \(g(r) \leq h(r)\) for all \(r\notin E\cup[0,1]\), where \(E\subset(1, +\infty)\) is a set of finite logarithmic measure. Let \(\alpha>1\) be a given constant. Then there exists \(r_{0}=r_{0}(\alpha)>0\) such that \(g(r) \leq h(\alpha r)\) for all \(r > r_{0}\).
Lemma 2.12
(see [6])
3 Proofs of Theorems 1.1-1.3
Proof of Theorem 1.1
Suppose that \(f(z)\) is a rational solution of (1.2). Since \(\sigma(A_{k})>\max\{\sigma(A_{j}), j=0,1,\ldots,k-1\}\), it is clear that \(f(z)\) is a polynomial with \(\deg f\leq k-1\).
Proof of Theorem 1.2
(i) Suppose that \(f(z)\) is a rational solution of (1.2). Since \(\sigma(A_{k})>\max\{\sigma(A_{j}), j=0,1,\ldots,k-1\}\), it is clear that \(f(z)\) is a polynomial with \(\deg f\leq k-1\).
(ii) Suppose that \(f(z)\) is a rational solution of (1.2). Since \(A_{k}(z)\) is transcendental and \(A_{j}(z)\), \(j=0,1,\ldots,k-1\), are polynomials, we can easily obtain that \(f(z)\) is a polynomial with \(\deg f\leq k-1\).
Proof of Theorem 1.3
We prove only the case under the hypotheses of Theorem 1.1, and the case under the hypotheses of Theorem 1.2 can be proved similarly. So, we omit the proof of the second case.
(i) Suppose that \(f(z)\) is a rational solution of (1.4). Since \(\sigma(A_{k})>\max\{\sigma(A_{j}), j=0, 1,\ldots, k-1, \sigma(F)\}\), it is clear that \(f(z)\) is a polynomial with \(\deg f\leq k-1\).
Since \(0<\varepsilon<\frac{\sigma-\delta}{2}\), (3.16) is a contradiction. Therefore, every transcendental meromorphic solution \(f(z)\) of (1.4), whose poles are of uniformly bounded multiplicities such that \(\lambda(\frac{1}{f})<\mu(f)\), satisfies \(\sigma(f)=\infty\).
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11301233, 11171119), the Youth Science Foundation of Education Bureau of Jiangxi Province (GJJ14271) and Sponsored Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University of China.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Hayman, WK: Meromorphic Functions. Clarendon, Oxford (1964) MATHGoogle Scholar
- Yang, L: Value Distribution Theory and Its New Research. Science Press, Beijing (1982) (in Chinese) Google Scholar
- Yang, CC, Yi, HX: Uniqueness Theory of Meromorphic Functions. Mathematics and Its Applications, vol. 557. Kluwer Academic, Dordrecht (2003) View ArticleMATHGoogle Scholar
- Cao, TB, Xu, JF, Chen, ZX: On the meromorphic solutions of linear differential equations on the complex plane. J. Math. Anal. Appl. 364(1), 130-142 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Kinnunen, L: Linear differential equations with solutions of finite iterated order. Southeast Asian Bull. Math. 22(4), 385-405 (1998) MATHMathSciNetGoogle Scholar
- Tu, J, Chen, ZX: Growth of solutions of complex differential equations with meromorphic coefficients of finite iterated order. Southeast Asian Bull. Math. 33, 153-164 (2009) MATHMathSciNetGoogle Scholar
- Belaïdi, B, Hamouda, S: Growth of solutions of an n-th order linear differential equation with entire coefficients. Kodai Math. J. 25, 240-245 (2002) View ArticleMathSciNetGoogle Scholar
- Belaïdi, B, Hamani, K: Order and hyper-order of entire solutions of linear differential equations with entire coefficients. Electron. J. Differ. Equ. 2003, 17 (2003) Google Scholar
- Chen, ZX, Gao, SA: The complex oscillation theory of certain non-homogeneous linear differential equations with transcendental entire coefficients. J. Math. Anal. Appl. 179, 403-416 (1993) View ArticleMATHMathSciNetGoogle Scholar
- Chen, ZX, Gao, SA: Entire solutions of differential equations with finite order transcendental entire coefficients. Acta Math. Sin. 13(4), 453-464 (1997) View ArticleMATHMathSciNetGoogle Scholar
- Chen, ZX: The growth of solutions of a type of second order differential equations with entire coefficients. Chin. Ann. Math., Ser. A 20(1), 7-14 (1999) (in Chinese) View ArticleMATHGoogle Scholar
- Chen, WJ, Xu, JF: Growth of meromorphic solutions of higher-order linear differential equations. Electron. J. Qual. Theory Differ. Equ. 2009, 1 (2009) View ArticleGoogle Scholar
- Lan, ST, Chen, ZX: On the growth of solutions of higher order differential equation. Acta Math. Appl. Sin. 36(5), 851-861 (2013) (in Chinese) MATHMathSciNetGoogle Scholar
- Hamani, K, Belaïdi, B: Growth of solutions of complex linear differential equations with entire coefficients of finite iterated order. Acta Univ. Apulensis 27, 203-216 (2011) MATHGoogle Scholar
- He, J, Zheng, XM, Hu, H: Iterated order of meromorphic solutions of certain higher order linear differential equations with meromorphic coefficients of finite iterated order. Acta Univ. Apulensis 33, 145-157 (2013) MathSciNetGoogle Scholar
- Hayman, WK, Rossi, JF: Characteristic, maximum modulus and value distribution. Trans. Am. Math. Soc. 284, 651-664 (1984) View ArticleMATHMathSciNetGoogle Scholar
- Gundersen, G: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 37(2), 88-104 (1988) MATHMathSciNetGoogle Scholar
- Barry, PD: On a theorem of Besicovitch. Q. J. Math. 14(1), 293-302 (1963) View ArticleMATHMathSciNetGoogle Scholar
- Barry, PD: Some theorems related to the \(\cos\pi\rho\) theorem. Proc. Lond. Math. Soc. 21(3), 334-360 (1970) View ArticleMATHMathSciNetGoogle Scholar
- Gundersen, G: Finite order solutions of second order linear differential equations. Trans. Am. Math. Soc. 305(1), 415-429 (1988) View ArticleMathSciNetGoogle Scholar