Oscillation of arbitrary-order derivatives of solutions to the higher order non-homogeneous linear differential equations taking small functions in the unit disc

1.   Introduction

In this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna's value distribution theory on the complex plane $\mathbb{C}$ and in the unit disc $\bigtriangleup = \{z\in \mathbb{C}:|z| < 1\}$ (see ^[1,2,3,4]). In addition, we need to give some definitions and discussions. Firstly, let us give two definitions about the degree of small growth order of functions in $\bigtriangleup$ as polynomials on the complex plane $\mathbb{C}$. There are many types of definitions of small growth order of functions in $\bigtriangleup$ (see ^[5,6]).

Definition 1.1. (see ^[5,6]). Let $f$ be a meromorphic function in $\bigtriangleup$, and

If $b < \infty$, then we say that $f$ is of finite $b$ degree (or is non-admissible). If $b = \infty$, then we say that $f$ is of infinite (or is admissible), both defined by characteristic function $T(r, f).$

Definition 1.2. (see ^[5,6]). Let $f$ be an analytic function in $\bigtriangleup$, and

Then we say that $f$ is a function of finite $a$ degree (or of infinite degree) defined by maximum modulus function $M(r, f) = \max\limits_{\mid z\mid = r}\mid f(z)\mid$.

Moreover, for $F\subset [0, 1)$, the upper and lower densities of $F$ are defined by

respectively, where $m(G) = \int_G\frac{dt}{1-t}$ for $G\subset[0, 1)$.

Now we give the definition of iterated order and growth index to classify generally the functions of fast growth in $\bigtriangleup$ as those in $\mathbb{C}$, see ^[3,7,8]. Let us define inductively, for $r\in[0, 1), \exp_1{r} = e^{r}$ and $\exp_{p+1}{r} = \exp(\exp_p{r}), p\in \mathbb{N}$. We also define for all $r$ sufficiently large in $(0, 1)$, $\log_1{r} = \log{r}$ and $\log_{p+1}{r} = \log(\log_p{r}), p\in \mathbb{N}$. Moreover, we denote by $\exp_0{r} = r, \log_0{r} = r, \exp_{-1}{r} = \log_1{r}, \log_{-1}{r} = \exp_1{r}$.

Definition 1.3. (see ^[9]). The iterated p-order of a meromorphic function $f$ in $\bigtriangleup$ is defined by

For an analytic function $f$ in $\bigtriangleup$, we also define

Remark 1.4. It follows by M. Tsuji in (^[4]) that if $f$ is an analytic function in $\bigtriangleup$, then

However it follows by (Proposition 2.2.2 in ^[3]) that

Definition 1.5. (see ^[9]). The growth index of the iterated order of a meromorphic function $f$ in $\bigtriangleup$ is defined by

For an analytic function $f$ in $\bigtriangleup$, we also define

Definition 1.6. (see ^[10,11]). Let $f$ be a meromorphic function in $\bigtriangleup$. Then the iterated p-exponent of convergence of the sequence of zeros of $f$ is defined by

where $N(r, \frac{1}{f})$ is the integrated counting function of zeros of $f(z)$ in $\{z\in \mathbb{C}:\mid z \mid < r\}$. Similarly, the iterated p-exponent of convergence of the sequence of distinct zeros of $f$ is defined by

where $\overline{N}(r, \frac{1}{f})$ is the integrated counting function of distinct zeros of $f$ in $\{z\in \mathbb{C}:\mid z \mid < r\}$.

Definition 1.7. (see ^[12]). Let $p\geq q\geq1$ be integers. Let $f$ be meromorphic function in $\bigtriangleup$, the $[p, q]$-order of $f$ is defined by

For an analytic function $f$ in $\bigtriangleup$, we also define

Remark 1.8. It is easy to see that $0\leq\rho_{[p, q]}(f)\leq \infty$. If $f$ is non-admissible, then $\rho_{[p, q]} = 0$ for any $p\geq q\geq 1$. By Definition 1.7, we have that $\rho_{[1,1]}(f) = \rho_{1}(f) = \rho(f),$ $ \rho_{^[2,1]}(f) = \rho_{2}(f) $and$ \rho_{[p+1, 1]}(f) = \rho_{p+1}(f) $.

Proposition 1.9. (see ^[12]). Let $p\geq q\geq 1$ be integers. Let f be analytic function in $\bigtriangleup$ of $[p, q]$-order. The following two statements hold:

(i) If $p = q$, then

(ii) If $p > q$, then

Definition 1.10. (see ^[13]). Let $p\geq q\geq 1$ be integers. The $[p, q]$-exponent of convergence of the zero sequence of a meromorphic function $f$ in $\bigtriangleup$ is defined by

Similarly, the $[p, q]$-exponent of convergence of the sequence of distinct zeros of $f$ is defined by

Definition 1.11. (see ^[1]). For $a\in \mathbb{\overline{C}} = \mathbb{C}\cup \{\infty\},$ the deficiency of $f$ is defined by

provided $f$ has unbounded characteristic.

The complex oscillation theory of solutions of linear differential equations in the complex plane $\mathbb{C}$ was started by S. Bank and I. Laine in 1982. Many authors have investigated the growth and oscillation of the solutions of complex linear differential equations in $\mathbb{C}$. In 2000, J. Heittokangas first studied the growth of the solution of linear differential equations in the unit disc $\bigtriangleup$. There already exist many results (see ^{[2,9,10,11,12,13]}) in $\bigtriangleup$, but the study is more difficult than that in $\mathbb{C}$, because the efficient tool, Wiman-Valiron theory, doesn't hold in $\bigtriangleup$. In 2015, author and L. P. Xiao (see ^[14]) studied the relationship between solutions and their derivatives of the differential equation

where $A(z), B(z)\not\equiv0$ and $F(z)\not\equiv0$ are meromorphic functions of finite iterated $p$-order in $\bigtriangleup$. Author obtained some oscillation theorems for $f^{(j)}(z)-\varphi(z)$, where $f$ is a solution and $\varphi(z)$ is a small function. Before we state author's results we need to define the following:

where $A_0(z) = A(z), B_0(z) = B(z)$ and $F_0(z) = F(z)$. Author and L. P. Xiao obtained the following results.

Theorem 1.1. (see ^[14]). Let $\varphi(z)$ be a meromorphic function in $\bigtriangleup$ with $\rho_p(\varphi) < \infty.$ Let $A(z)$, $B(z)\not\equiv0$ and $F(z)\not\equiv0$ be meromorphic functions of finite iterated p-order in $\bigtriangleup$ such that $B_j(z)\not\equiv0$ and $D_j(z)\not\equiv0$ $(j = 0, 1, 2, \cdots).$

(i) If $f$ is a meromorphic solution in $\bigtriangleup$ of (1.1) with $\rho_p(f) = \infty$ and $\rho_{p+1}(f) = \rho < \infty$, then $f$ satisfies

(ii) If $f$ is a meromorphic solution in $\bigtriangleup$ of (1.1) with

then

Theorem 1.2. (see ^[14]). Let $\varphi(z)$ be an analytic function in $\bigtriangleup$ with $\rho_p(\varphi) < \infty$ and be not a solution of (1.1). Let $A(z)$, $B(z)\not\equiv0$ and $F(z)\not\equiv0$ be analytic functions in $\bigtriangleup$ with finite iterated p-order such that $\beta = \rho_p(B) > \max\{\rho_p(A), \rho_p(F), \rho_p(\varphi)\}$ and $\rho_{M, p}(A)\leq\rho_{M, p}(B).$ Then all nontrivial solutions of (1.1) satisfy

with at most one possible exceptional solution $f_0$ such that

Theorem 1.3. (see ^[14]). Let $\varphi(z)$ be a meromorphic function in $\bigtriangleup$ with $\rho_p(\varphi) < \infty$ and be not a solution of (1.1). Let $A(z)$, $B(z)\not\equiv0$ and $F(z)\not\equiv0$ be meromorphic functions in $\bigtriangleup$ with finite iterated p-order such that $\rho_p(B) > \max\{\rho_p(A), \rho_p(F), \rho_p(\varphi)\}$ and $\delta(\infty, B) > 0.$ If $f$ is a meromorphic solution in $\bigtriangleup$ of (1.1) with $\rho_p(f) = \infty$ and $\rho_{p+1}(f) = \rho$, then $f$ satisfies

In 2018, Z. Dahmani and M. A. Abdelaoui (see ^[15]) studied the higher order non-homogeneous linear differential equation

where $A_{j}(z)(j = 0, 1, \cdots, k-1)$, and $F(z)\not\equiv0$ are meromorphic functions of finite iterated $[p, q]$-order in $\bigtriangleup$. Before we state their results we need to define the following:

where $\Psi^{i-1}_{j+1} = \frac{A^{i-1}_{j+1}}{A^{i-1}_{0}}$. Z. Dahmani and M. A. Abdelaoui obtained the following results.

Theorem 1.4. (see ^[15]) Let $p\geq q\geq1$ be integers, and let $A_{j}(z)(j = 0, 1, \cdots, k-1), F(z)\not\equiv0$ and $\varphi(z)$ be meromorphic functions in $\bigtriangleup$ of finite $[p, q]$-order such that $D_{i}(z)\not\equiv0$ $(i = 0, 1, 2, \cdots)$. If $f$ is a meromorphic solution of the Eq (1.6) of infinite $[p, q]$-order and $\rho_{[p+1, q]}(f) = \rho$, then $f$ satisfies

Theorem 1.5. (see ^[15]). Let $p\geq q\geq1$ be integers, and let $A_{j}(z)(j = 0, 1, \cdots, k-1), F(z)\not\equiv0$ and $\varphi(z)$ be meromorphic functions in $\bigtriangleup$ of finite $[p, q]$-order such that $D_{i}(z)\not\equiv0$ $(i = 0, 1, 2, \cdots)$. If $f$ is a meromorphic solution of the Eq (1.6) with

then $f$ satisfies

2.   Main results

According to the proof process of Theorem 1.4 and Theorem 1.5, we know that it is necessary to increase the condition $A^{i}_{0}(z)\not\equiv0$ and $D_{i}(z)\not\equiv0$ $(i = 0, 1, 2, \cdots)$ to ensure that the Theorem 1.4 and the Theorem 1.5 are established, because we need to divide both sides of the higher order non-homogeneous linear differential equations by $A^{i}_{0}(z)$. Where $A^{i}_{0}(z)$ and $D_{i}(z)$ are defined in (1.7), (1.9) and (1.11). In this article, we give some sufficient conditions on the coefficients which guarantee $A^{i}_{0}(z)\not\equiv0$ and $D_{i}(z)\not\equiv0$ $(i = 0, 1, 2, \cdots)$, and we obtain:

Theorem 2.1. Let $p\geq q\geq1$ be integers, and let $\varphi(z)$ be an analytic function in $\bigtriangleup$ with $\rho_{[p, q]}(\varphi) < \infty$ and be not a solution of (1.6). Let $A_{j}(z)(j = 1, 2, \cdots, k-1)$, $A_{0}(z)\not\equiv0$ and $F(z)\not\equiv0$ be analytic functions in $\bigtriangleup$ of finite $[p, q]$-order such that $\beta = \rho_{[p, q]}(A_{0}) > \max\{\rho_{[p, q]}(A_{j}) (j = 1, 2, \cdots, k-1), \rho_{[p, q]}(F), \rho_{[p, q]}(\varphi)\}$ and $\rho_{M, [p, q]}(A_{j})\leq \rho_{M, [p, q]}(A_{0})$ $(j = 1, 2, \cdots, k-1)$. Then all nontrivial solutions of (1.6) satisfy

with at most one possible exceptional solution $f_0$ such that

Theorem 2.2. Let $p\geq q\geq1$ be integers, and let $\varphi(z)$ be an meromorphic function in $\bigtriangleup$ with $\rho_{[p, q]}(\varphi) < \infty$ and be not a solution of (1.6). Let $A_{j}(z)(j = 1, 2, \cdots, k-1)$, $A_{0}(z)\not\equiv0$ and $F(z)\not\equiv0$ be meromorphic functions in $\bigtriangleup$ of finite $[p, q]$-order such that $\rho_{[p, q]}(A_{0}) > \max\{\rho_{[p, q]}(A_{j}) (j = 1, 2, \cdots, k-1), \rho_{[p, q]}(F), \rho_{[p, q]}(\varphi)\}$ and $\delta(\infty, A_{0}) > 0$. If $f$ is a meromorphic solution in $\bigtriangleup$ of (1.6) with $\rho_{[p, q]}(f) = \infty$ and $\rho_{[p+1, q]}(f) = \rho$, then $f$ satisfies

3.   Some lemmas

To prove our theorems, we require the following lemmas.

Lemma 3.1. (see ^[13]). Let $p\geq q\geq1$ be integers, and let $A_{0}, A_{1}, \cdots, A_{k-1}$ be analytic functions in $\bigtriangleup$ satisfying

If $f\not\equiv0$ is a solution of (3.1), then $\rho_{[p, q]}(f) = \infty$ and

Furthermore, if $p > q$, then

Lemma 3.2. (see ^[15]). Let $p\geq q\geq1$ be integers. Let $A_{0}, A_{1}, \cdots, A_{k-1}$ and $F\not\equiv0$ be meromorphic functions in $\bigtriangleup$ and let $f$ be a meromorphic solution of (1.6) satisfying $\max\{\rho_{[p, q]}(A_{j}) (j = 0, 1, 2, \cdots, k-1), \rho_{[p, q]}(F)\} < \rho_{[p, q]}(f)\leq \infty,$ then we have

Lemma 3.3. Let $p\geq q\geq1$ be integers, and assume that coefficients $A_{0}, A_{1}, \cdots, A_{k-1}$ and $F\not\equiv0$ are analytic in $\bigtriangleup$ and $\rho_{[p, q]}(A_{j}) < \rho_{[p, q]}(A_{0})$ for all $j = 1, 2, \cdots, k-1$. Let $\alpha_{M} = \max\{\rho_{M, [p, q]}(A_{j}):j = 0, 1, \cdots, k-1\}$. If $\rho_{M, [p+1, q]}(F) < \rho_{[p, q]}(A_{0})$, then all solutions $f$ of (1.6) satisfy

with at most one exceptional $f_0$ satisfying $\rho_{M, [p+1, q]}(f_0) < \rho_{[p, q]}(A_{0})$.

$Proof$. Let $f_{1}, f_{2}, \cdots, f_{k}$ be a solution base of the differential equation

Then by the elementary theory of differential equations (see ^[3]), any solution of (1.6) can be represented in the form

where $C_{1}, C_{2}, \cdots, C_{k}\in \mathbb{C}$ and $B_{1}, B_{2}, \cdots, B_{k}$ are analytic in $\bigtriangleup$ given by the system of equations

Since the Wronskian of $f_{1}, f_{2}, \cdots, f_{k}$ satisfies $W(f_{1}, f_{2}, \cdots, f_{k}) = \exp(-\int A_{k-1}dz)$, we obtain

where $G_{j}(f_{1}, f_{2}, \cdots, f_{k})$ is a differential polynomial of $f_{1}, f_{2}, \cdots, f_{k}$ and of their derivatives, with constant coefficients. Then by Lemma 3.1, we know that $\alpha_{M}\geq \rho_{M, [p+1, q]}(f_{j})\geq \rho_{[p, q]}(A_{0})$. By (3.2)–(3.4), we have

Since $\rho_{M, [p+1, q]}(F) < \rho_{[p, q]}(A_{0})\leq \alpha_{M}$, it follows from (3.5) and (1.6) that all solutions $f$ of (1.6) satisfy $\rho_{M, [p+1, q]}(f)\leq \alpha_{M}$.

Now we assert that all solutions $f$ of (1.6) satisfy $\rho_{M, [p+1, q]}(f)\geq \rho_{[p, q]}(A_{0})$ with at most one exception. In fact, if there exist two distinct solutions $g_{1}, g_{2}$ of (1.6) with $\rho_{M, [p+1, q]}(g_{i}) < \rho_{[p, q]}(A_{0})$ $(i = 1, 2)$, then $g = g_{1}-g_{2}$ satisfies $\rho_{M, [p+1, q]}(g) = \rho_{M, [p+1, q]}(g_{1}-g_{2}) < \rho_{[p, q]}(A_{0})$. But $g$ is a nonzero solution of (3.1) satisfying $\rho_{M, [p+1, q]}(g) = \rho_{M, [p+1, q]}(g_{1}-g_{2})\geq\rho_{[p, q]}(A_{0})$ by Lemma 3.1. This is a contradiction.

By Lemma 3.2, all solutions $f$ of (1.6) satisfy $\alpha_{M}\geq\rho_{M, [p+1, q]}(f) = \overline{\lambda}_{[p+1, q]}(f) = \lambda_{[p+1, q]}(f)\geq \rho_{[p, q]}(A_{0})$, with at most one exceptional $f_0$ satisfying $\rho_{M, [p+1, q]}(f_0) < \rho_{[p, q]}(A_{0})$.

Lemma 3.4. Let $p\geq q\geq1$ be integers, $\varphi$ be finite $[p, q]$-order analytic functions in $\bigtriangleup$ and assume that coefficients $A_{0}, A_{1}, \cdots, A_{k-1}$, $F\not\equiv0$ and $F-\varphi^{(k)}-A_{k-1}\varphi^{(k-1)}-\cdots-A_{1}\varphi'-A_{0}\varphi \not\equiv 0$ are analytic in $\bigtriangleup$ and $\rho_{[p, q]}(A_{j}) < \rho_{[p, q]}(A_{0})$ for all $j = 1, 2, \cdots, k-1$. Let $\alpha_{M} = \max\{\rho_{M, [p, q]}(A_{j}):j = 0, 1, \cdots, k-1\}$. If $\rho_{M, [p+1, q]}(F-\varphi^{(k)}-A_{k-1}\varphi^{(k-1)}-\cdots-A_{1}\varphi'-A_{0}\varphi) < \rho_{[p, q]}(A_{0})$, then all solutions $f$ of (1.6) satisfy

with at most one exceptional $f_0$ satisfying $\rho_{M, [p+1, q]}(f_0) < \rho_{[p, q]}(A_{0})$.

$Proof$. Suppose that $g = f-\varphi$, obtain $f = g+\varphi$, then from (1.6) we have $g^{(k)}+A_{k-1}g^{(k-1)}+\cdots+A_{1}g'+A_{0}g = F-\varphi^{(k)}-A_{k-1}\varphi^{(k-1)}-\cdots-A_{1}\varphi'-A_{0}\varphi$. By Lemma 3.3 we obtain all solutions $f$ of (1.6) satisfy

with at most one exceptional $f_0$ satisfying $\rho_{M, [p+1, q]}(f_0) < \rho_{[p, q]}(A_{0})$.

Lemma 3.5. (see ^[12]). Let $p\geq q\geq1$ be integers. Let $f$ be a meromorphic function in $\bigtriangleup$ such that $\rho_{[p, q]}(f) = \rho < \infty,$ and let $k\geq1$ be an integer. Then for any $\varepsilon > 0$,

holds for all $r$ outside a set $E_1 \subset[0, 1)$ with $\int_{E_{1}} < /italic > < italic > \frac{dr}{1-r} < \infty.$

4.   Proofs of Theorems 2.1 and 2.2

4.1.   The proof of Theorem 2.1

Since $F-(\varphi^{(k)}+A_{k-1}\varphi^{(k-1)}+\cdots+A_{1}\varphi'+A_{0}\varphi)\not\equiv 0$, $\rho_{M, [p+1, q]}(F-(\varphi^{(k)}+A_{k-1}\varphi^{(k-1)}+\cdots+A_{1}\varphi'+A_{0}\varphi)) < \rho_{[p, q]}(A_{0})$. By Lemma 3.4, all nontrivial solutions of (1.6) satisfy

with at most one exceptional $f_0$ such that $\rho_{[p+1, q]}(f_0) < \rho_{[p, q]}(A_{0})$. By using (1.9) we have

Now we prove that $A^{i}_{0}\not\equiv 0$ for all $i = 1, 2, 3, \cdots$. For that we suppose there exists $i\in \mathbb{N}$ such that $A^{i}_{0} = 0$. By (4.1) and Lemma 3.5 we have for any $\varepsilon > 0$,

outside a set $E_1 \subset[0, 1)$ with $\int_{E_{1}} \frac{dr}{1-r} < \infty,$ for all $i = 1, 2, 3, \cdots$, $\beta = \rho_{[p, q]}(A_{0})$. Which implies the contradiction

Hence $A^{i}_{0}\not\equiv 0$ for all $i = 1, 2, 3, \cdots$. We prove that $D_{i}\not\equiv 0$ for all $i = 1, 2, 3, \cdots$. For that we suppose there exists $i\in \mathbb{N}$ such that $D_{i} = 0$. We have $F_{i}-(\varphi^{(k)}+A^{i}_{k-1}\varphi^{(k-1)}+\cdots+A^{i}_{0}\varphi) = 0$ from (1.11), which implies

Here we suppose that $\varphi(z)\not\equiv 0$,

On the other hand, from (1.10),

By (4.3), (4.4) and Lemma 3.5 we have

which implies the contradiction

If $\varphi(z)\equiv 0$, then from (1.10) and (1.11)

which implies $F_{i-1}(z) = cA^{i-1}_{0}(z)$, where $c$ is some constant. By (4.1) and (4.6), we have

By (4.4), (4.7) and Lemma 3.5 we have

which implies the contradiction

Hence $D_{i}\not\equiv 0$ for all $i = 1, 2, 3, \cdots$. Since $A^{i}_{0}\not \equiv0$, $D_{i}\not\equiv0$ $(i = 1, 2, 3, \cdots)$, then by Theorem 1.4 and Lemma 3.4 we have

with at most one possible exceptional solution $f_0$ such that

Therefore, the proof of Theorem 2.1 is completely.

4.2.   The proof of Theorem 2.2

We need only to prove that $A^{i}_{0}\not\equiv0$ and $D_{i}\not\equiv0$ for all $j = 1, 2, 3, \cdots$. Then by Theorem 1.4 we can obtain Theorem 2.2. Consider the assumption $\delta(\infty, A_{0}) = \delta > 0$. Then for $r\rightarrow1^-$ we have

Now we prove that $A^{i}_{0}\not\equiv 0$ for all $i = 1, 2, 3, \cdots$. For that we suppose there exists $i\in \mathbb{N}$ such that $A^{i}_{0} = 0$. By (4.1) and (4.8) we obtain

which implies the contradiction

Hence $A^{i}_{0}\not\equiv 0$ for all $i = 1, 2, 3, \cdots$. We prove that $D_{i}\not\equiv 0$ for all $i = 1, 2, 3, \cdots$. For that we suppose there exists $i\in \mathbb{N}$ such that $D_{i} = 0$. If $\varphi(z)\not\equiv 0$, then by (4.3), (4.4), (4.8) and Lemma 3.5 we have

which implies the contradiction

If $\varphi(z)\equiv 0$, then by (4.4), (4.7) and Lemma 3.5 we have

which implies the contradiction

Hence $D_{i}\not\equiv 0$ for all $i = 1, 2, 3, \cdots$. By Theorem 1.4, we have Theorem 2.2.

Therefore, this completes the proof of Theorem 2.2.

5.   Conclusions

We first obtained some oscillation theorems (see ^[14]) which consider the distribution of meromorphic solutions and their arbitrary-order derivatives taking small function values instead of taking zeros. Moreover, Z. Dahmani and M. A. Abdelaoui (see ^[15]) investigated the higher order non-homogeneous linear differential equation which can be seen as an improvement of ^[14]. By using those theorems, we obtain some oscillation theorems for $f^{(j)}(z)-\varphi(z)$, where $f$ is a solution and $\varphi(z)$ is a small function. We believe our results will attract the attentions of the related readers.

Acknowledgments

The authors would like to thank the anonymous referee for making valuable suggestions and comments to improve this article.

This work was supported by the National Natural Science Foundation of China (12161074), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001), and the Foundation of Education Department of Jiangxi (GJJ190895, GJJ190876) of China.

Conflict of interest

The authors declare that none of the authors have any competing interests in the manuscript.