On hyper-order of solutions of higher order linear differential equations with meromorphic coefficients
- Jianren Long^{1}Email author and
- Jun Zhu^{1}
https://doi.org/10.1186/s13662-016-0806-6
© Long and Zhu 2016
Received: 25 October 2015
Accepted: 8 March 2016
Published: 15 April 2016
Abstract
Keywords
complex differential equation meromorphic function hyper-orderMSC
34M10 30D351 Introduction and main results
Theorem 1.1
([8])
In 2000, Chen and Yang studied the hyper-order of solutions of (1.1).
Theorem 1.2
([9])
In Theorems 1.1 and 1.2, the authors consider all coefficients are entire functions. When the coefficients \(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z)\) and \(F(z)\) are meromorphic functions, many authors investigated the value distribution of solutions of (1.1) and (1.3); see, for example, [10–18]. Especially, we mention the following result given by Chen [14], in which a precise estimation of hyper-order of solutions of (1.1) is obtained.
Theorem 1.3
([14], Theorem 2)
In 2005, Xiao and Chen considered the non-homogeneous equation (1.3), the following result is proved.
Theorem 1.4
([19])
Belaïdi studied equation (1.1), a precise estimation of hyper-order of solutions of (1.1) is also obtained by using different conditions from those mentioned above, in which the growths of the coefficients are limited in a set having positive densities.
Theorem 1.5
([20])
Theorem 1.5 and the remaining theorems involve the logarithmic measure and densities of set, which will be recalled in Section 2. In this paper, we study the growth of solutions of (1.1) and (1.3), and one of the goals is to extend Theorems 1.3 and 1.4 in which the condition \(\rho(A_{s})<\frac{1}{2}\) is deleted. On the other hand, we consider the case of a meromorphic coefficient in Theorem 1.5. The following results are proved by combining the methods of Theorems 1.3, 1.4, and 1.5.
Theorem 1.6
For the case of non-homogeneous equation, we get the following result.
Theorem 1.7
2 Auxiliary results
A lemma on logarithmic derivatives due to Gundersen [21] plays an important role in proving our results.
Lemma 2.1
The following result was proved originally in [22]; see also [14], Lemma 3.
Lemma 2.2
The next lemma is related to the central index.
Lemma 2.3
([14], Lemma 2)
Lemma 2.4
Let \(f(z)=\frac{g(z)}{d(z)}\) be a meromorphic function, where \(g(z)\) and \(d(z)\) are entire functions. If \(0\leq\rho(d)<\mu(f)\), then \(\mu (g)=\mu(f)\), \(\rho(g)=\rho(f)\). Moreover, if \(\rho(f)=\infty\), then \(\rho_{2}(f)=\rho_{2}(g)\).
Proof
We divide the proof into the following three cases.
Case 3. \(\mu(f)<\infty\) and \(\rho(f)=\infty\). In a similar way to proving cases 1 and 2, we can prove case 3.
Lemma 2.5
([23], Lemma 2)
Lemma 2.6
([5], Lemma 5)
Let \(g: [0, \infty)\rightarrow\mathbf{R}\) and \(h: [0, \infty )\rightarrow\mathbf{R}\) be monotonically nondecreasing functions such that \(g(r)\leq h(r)\) outside of an exceptional set E with \(\mathrm{m}_{\mathrm{l}}(E)<\infty\). Then, for any \(\alpha>1\), there exists an \(r_{0}>1\) such that \(g(r)\leq h(\alpha r)\) for all \(r>r_{0}\).
Lemma 2.7
Proof
Lemma 2.8
Proof
In a similar way to proving Lemma 2.7, we can get the proof of Lemma 2.8, here we omit the details. □
Lemma 2.9
Proof
Lemma 2.10
Proof
3 Proof of Theorems 1.6 and 1.7
Proof of Theorem 1.6
Proof of Theorem 1.7
Declarations
Acknowledgements
The authors thank the referee for his/her valuable suggestions to improve the paper. This research work is supported by the Foundation of Science and Technology of Guizhou Province of China (Grant No. [2015]2112), the National Natural Science Foundation of China (Grant No. 11501142).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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