# On hyper-order of solutions of higher order linear differential equations with meromorphic coefficients

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## Abstract

*etc.*

### Keywords

complex differential equation meromorphic function hyper-order### MSC

34M10 30D35## 1 Introduction and main results

*f*in the complex plane \(\mathbb{C}\), the order of growth and the lower order of growth are defined as

*f*is defined as

*et al.*in [8].

### Theorem 1.1

([8])

*Let*\(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z)\), \(F(z)\)

*be entire functions*.

*Suppose there exists an integer*

*s*, \(0\leq s\leq k-1\),

*such that*

*Then every solution of*

*is either a polynomial or an entire function of infinite order*.

In 2000, Chen and Yang studied the hyper-order of solutions of (1.1).

### Theorem 1.2

([9])

*Let*\(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z)\)

*be entire function satisfying*

*Then every nontrivial solution*

*f*

*of*(1.1)

*satisfies*\(\rho _{2}(f)=\rho(A_{0})\).

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, 11, 12, 13, 14, 15, 16, 17, 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)

*Let*\(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z)\)

*be meromorphic functions*.

*Suppose there exists an integer*

*s*, \(0\leq s\leq k-1\),

*satisfying*

*If equation*(1.1)

*has a meromorphic solution*,

*then every transcendental meromorphic solution*

*f*

*of*(1.1)

*satisfies*\(\rho _{2}(f)=\rho(A_{s})\).

In 2005, Xiao and Chen considered the non-homogeneous equation (1.3), the following result is proved.

### Theorem 1.4

([19])

*Let*\(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z), F(z)\)

*be meromorphic functions*.

*Suppose there exists an integer*

*s*, \(0\leq s\leq k-1\),

*satisfying*

*If equation*(1.3)

*has a meromorphic solution*,

*then every transcendental meromorphic solution*

*f*

*of*(1.3)

*satisfies*\(\rho _{2}(f)=\rho(A_{s})\).

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])

*Let*

*E*

*be a set of complex numbers satisfying*\(\overline{\operatorname{dens}}(\{|z|: z\in E\})>0\),

*and let*\(A_{j}(z)\), \(j=0, 1, \ldots, k-1\),

*be entire functions such that*

*and for some constants*\(0\leq\beta<\alpha\),

*we have*,

*for all*\(\varepsilon>0\)

*sufficiently small*,

*as*\(z\rightarrow\infty\)

*for*\(z\in E\).

*Then every nontrivial solution*

*f*

*of*(1.1)

*satisfies*\(\rho _{2}(f)=\rho(A_{0})\).

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

*Let*

*E*

*be a set of complex numbers satisfying*\(\mathrm{m}_{\mathrm{l}}(\{ |z|: z\in E\})=\infty\),

*and let*\(A_{j}(z)\), \(j=0, 1, \ldots, k-1\),

*be meromorphic functions*.

*Suppose there exists an integer*

*s*, \(0\leq s\leq k-1\),

*satisfying*

*and for some constants*\(0\leq\beta<\alpha\),

*we have*,

*for all*\(\varepsilon>0\)

*sufficiently small*,

*as*\(z\rightarrow\infty\)

*for*\(z\in E\).

*Then every nontrivial meromorphic solution*

*f*

*whose poles are of uniformly bounded multiplicities of equation*(1.1)

*satisfies*\(\rho_{2}(f)=\rho(A_{s})\).

For the case of non-homogeneous equation, we get the following result.

### Theorem 1.7

*Let*

*E*

*and*\(A_{j}(z)\), \(j=0, 1, \ldots, k-1\)

*be defined as Theorem*1.6,

*and let*\(F(z)\not\equiv0\)

*be meromorphic function*.

*Suppose there exists an integer*

*s*, \(0\leq s\leq k-1\),

*satisfying*

*and for some constants*\(0\leq\beta<\alpha\),

*we have*,

*for all*\(\varepsilon>0\)

*sufficiently small*,

*equations*(1.4)

*and*(1.5)

*hold as*\(z\rightarrow\infty\)

*for*\(z\in E\).

*Then every nontrivial meromorphic solution*

*f*

*whose poles are of uniformly bounded multiplicities of equation*(1.3)

*satisfies*\(\rho_{2}(f)=\rho(A_{s})\).

## 2 Auxiliary results

A lemma on logarithmic derivatives due to Gundersen [21] plays an important role in proving our results.

### Lemma 2.1

*Let*

*f*

*be a transcendental meromorphic function*,

*and let*\(\alpha>1\)

*be a given real constant*.

*Let*

*k*

*and*

*j*

*be two integers such that*\(k>j\geq0\).

*Then there exists a set*\(E\subset(1, \infty)\)

*with*\(\mathrm{m}_{\mathrm{l}}(E)<\infty\),

*and a constant*\(B>0\)

*depending only on*

*α*

*and*

*j*,

*k*,

*such that*,

*for all*

*z*

*satisfying*\(|z|\notin(E\cup[0, 1])\),

*we have*

The following result was proved originally in [22]; see also [14], Lemma 3.

### Lemma 2.2

*Let*

*f*

*be a meromorphic function of order*\(\rho(f)=\beta<\infty\).

*Then*,

*for any given*\(\varepsilon>0\),

*there exists a set*\(E\subset(1, \infty)\)

*with*\(\mathrm{m}_{\mathrm{l}}(E)<\infty\)

*and*\(\mathrm{m} (E)<\infty\),

*such that*,

*for all*

*z*

*satisfying*\(|z|=r\notin([0, 1]\cup E)\),

The next lemma is related to the central index.

### Lemma 2.3

([14], Lemma 2)

*Let*\(f(z)=\frac{g(z)}{d(z)}\)

*be a meromorphic function*,

*where*\(g(z)\)

*and*\(d(z)\)

*are entire functions satisfying*

*Then there exists a set*\(E\subset(1, \infty)\)

*with*\(\mathrm{m}_{\mathrm{l}} (E)<\infty\),

*such that*,

*for all*

*z*

*satisfying*\(|z|=r\notin ([0, 1]\cup E)\)

*and*\(|g(z)|=M(r, g)\), \(M(r, g)=\max_{|z|=r}|g(z)|\),

*we have*

*where*\(\nu_{g}(r)\)

*denotes the central index of*\(g(z)\).

### 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.

*n*. This implies that, for all sufficiently large

*n*,

*n*,

*n*. Then, for all sufficiently large

*n*,

Case 3. \(\mu(f)<\infty\) and \(\rho(f)=\infty\). In a similar way to proving cases 1 and 2, we can prove case 3.

*N*, such that, for \(n>N\),

### Lemma 2.5

([23], Lemma 2)

*Let*

*f*

*be an entire function of infinite order*,

*with the hyper*-

*order*\(\rho_{2}(f)<\infty\),

*and let*\(\nu(r)\)

*be the central index of*

*f*.

*Then*

### 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

*Let*

*f*

*be a meromorphic solution of equation*(1.1),

*where*\(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z)\)

*are meromorphic functions*.

*If there exists an integer number*\(s\in\{0, 1, 2, \ldots, k-1\}\)

*that satisfies*

*then*\(\rho(f)\geq\rho(A_{s})\), \(\mu(f)\geq\mu(A_{s})\).

### Proof

### Lemma 2.8

*Let*

*f*

*be a meromorphic solution of equation*(1.3),

*where*\(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z), F(z)\not\equiv0\)

*are meromorphic functions*.

*If there exists an integer number*\(s\in\{0, 1, 2, \ldots, k-1\}\)

*that satisfies*

*then*\(\rho(f)\geq\rho(A_{s})\), \(\mu(f)\geq\mu(A_{s})\).

### 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

*Let*\(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z)\)

*be meromorphic functions of finite order*.

*Suppose there exists an integer number*\(s\in\{0, 1, 2, \ldots, k-1\}\),

*such that*

*Then every infinite order meromorphic solution*

*f*

*whose poles are of uniformly bounded multiplicities of equation*(1.1)

*satisfies*\(\rho_{2}(f)\leq\rho(A_{s})\).

### Proof

*z*satisfying \(|z|=r\notin([0, 1]\cup E_{1})\),

*f*come from the poles of \(A_{j}(z)\), \(j=0, 1, \ldots, k-1\), and the multiplicities of poles of

*f*are uniformly bounded, we have \(\lambda(\frac{1}{f})<\mu(A_{s})\). Set \(f(z)=\frac {g(z)}{d(z)}\), where \(g(z)\) is an entire function, \(d(z)\) is the classic product of poles sequence of

*f*. It follows from Lemmas 2.4 and 2.7 that

*z*satisfying \(|z|=r\notin([0, 1]\cup E_{2})\), \(|g(z)|=M(r, g)\),

*z*satisfying \(|z|=r\notin([0, R]\cup E_{1}\cup E_{2})\), \(|g(z)|=M(r, g)\), and \(\nu _{g}(r)>1\),

### Lemma 2.10

*Let*\(A_{0}(z), A_{1}(z), \ldots, A_{k-1}(z)\)

*and*\(F(z)\not\equiv0\)

*be meromorphic functions of finite order*.

*Suppose there exists an integer number*\(s\in\{0, 1, 2, \ldots, k-1\}\),

*such that*

*Then every infinite order meromorphic solution*

*f*

*whose poles are of uniformly bounded multiplicities of equation*(1.3)

*satisfies*\(\rho_{2}(f)\leq\rho(A_{s})\).

### Proof

*z*satisfying \(|z|=r\notin([0, 1]\cup E_{3})\), we have (2.3) and

*f*come from the poles of \(A_{j}(z)\), \(j=0, 1, \ldots, k-1\), \(F(z)\), and the multiplicities of poles of

*f*are uniformly bounded, we have \(\lambda(\frac{1}{f})<\mu(A_{s})\). Set \(f(z)=\frac{g(z)}{d(z)}\), where \(g(z)\) is an entire function, \(d(z)\) is a classic product of poles sequence of

*f*. It follows from Lemmas 2.3, 2.4 and 2.8 that there exists a set \(E_{4}\subset(1, \infty)\) with \(\mathrm{m}_{\mathrm{l}}(E_{4})<\infty\), such that, for all

*z*satisfying \(|z|=r\notin([0, 1]\cup E_{4})\), \(|g(z)|=M(r, g)\), equation (2.4) holds.

*z*satisfying \(|z|=r\notin([0, R]\cup E_{3}\cup E_{4})\), \(|g(z)|=M(r, g)\),

## 3 Proof of Theorems 1.6 and 1.7

### Proof of Theorem 1.6

*z*satisfying \(|z|=r\notin([0, R_{1}]\cup E_{1})\), where \(R_{1}>1\) is a constant,

*z*satisfying \(|z|=r\notin([0, 1]\cup E_{2})\) and \(|g(z)|=M(r, g)\),

*z*satisfying \(|z|=r>R_{2}\), \(\nu _{g}(r)>1\), \(|1+o(1)|>\frac{1}{2}\), and \(|g(z)|=M(r, g)\), \(M(r, g)>1\),

*z*satisfying \(|z|=r\in H_{1}\) and \(|g(z)|=M(r, g)\),

### Proof of Theorem 1.7

*z*satisfying \(|z|=r\notin([0, R_{3}]\cup E_{3})\), where \(R_{3}>1\) is a constant, equations (3.2) and (3.3) hold.

*f*. Since the poles of

*f*come from the poles of \(A_{j}(z)\), \(j=0, 1, \ldots, k-1\), \(F(z)\), and the multiplicities of the poles of

*f*are uniformly bounded, we have \(\lambda(\frac{1}{f})=\lambda(d)=\rho(d)\leq b\), where

*η*be constant satisfying \(b<\eta<\mu(A_{s})\). By Lemma 2.2, there exists a set \(E_{4}\subset(1, \infty)\) with \(\mathrm{m}_{\mathrm{l}}(E_{4})<\infty\), such that, for all

*z*satisfying \(|z|=r\notin([0, 1]\cup E_{4})\),

*z*satisfying \(|z|=r\notin([0, 1]\cup E_{5})\) and \(|g(z)|=M(r, g)\), equation (3.4) holds. Therefore, there exists a constant \(R_{4}\) (\(>R_{3}\)), such that, for all

*z*satisfying \(|z|=r>R_{4}\), \(\nu_{g}(r)>1\), \(|1+o(1)|>\frac{1}{2}\), and \(|g(z)|=M(r, g)\), \(M(r, g)>1\), equation (3.5) holds.

*z*satisfying \(|z|=r\in H_{2}\), \(r>R_{4}\), and \(|g(z)|=M(r, g)\),

*z*satisfying \(|z|=r\in H_{2}\) and \(|g(z)|=M(r, g)\),

## Notes

### 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).

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