On properties of solutions of complex differential equations in the unit disc

1.   Introduction

In 1982, Pommerenke studied the properties of solutions of second order differential equation

where $A(z)$ is an analytic function in $\mathbb{D} = \{z:|z| < 1\}$, some conditions of $A(z)$ such that every solution of (1.1) belong to the Hardy space $H^{2}$ have been obtained in ^[15]. The concepts concerning Hardy space and other related spaces will be given below. Later on, Heittokangas investigated the properties of solutions of higher order differential equation

where $A(z)$ is analytic function in $\mathbb{D}$, some conditions of the coefficient $A(z)$ such that all solutions of (1.2) belong to some analytic spaces, for example, weighted Hardy space, Bloch space and so on, see ^[7] for more details. At the same time, the following equation

was also studied, where $A_{j}(z)$ and $F(z)$ are analytic in $\mathbb{D}$, $j = 0, 1, \ldots, k-1$, $k\geq 2$. It was shown that all solutions of (1.3) are analytic in $\mathbb{D}$, for more details refers to [7,Theorem 7.1,p. 42]. From that time, it has been very interesting to investigate function space's properties of solutions of linear differential equations in $\mathbb{D}$, and more and more results have been obtained by many different researchers, for example, see ^{[5,8,9,12,14,16]} and references therein. Here the properties of solutions of (1.3) are studied again, in which the growth of solutions and function space's properties of solutions are considered. According to the function space's properties, usually, we consider two kind questions, one is called direct problem, in which we ask the properties of solution by using condition of coefficients. Another aspect is called the inverse problem, in which we ask the properties of coefficients by using the conditions of solutions.

In order to state our results, some concepts will be recalled. Let $H(\mathbb{D})$ denotes a set of all holomorphic functions in $\mathbb{D}$. Let $0 < p < \infty$, the Hardy space $H^{p}$ is defined as

see ^[4] for more details. For the case of $p = \infty$, $f(z)\in{H(\mathbb{D})}$ is said to belong to $H^{\infty}$ if and only if

Obviously, $H^{\infty}$ denotes the set of bounded analytic functions in $\mathbb{D}$. For $0\leq q < \infty$, the weighted Hardy space $H_{q}^{\infty}$ is defined as

Obviously, $H_{0}^{\infty} = H^{\infty}$. Moreover, $f(z)$ is said to belong to $G_{p}$ if

more details of $G_{p}$ can be found in ^[2].

The Bloch space is defined as

For $\alpha > 0$, the $\alpha$-Bloch space is also defined by

which can be found in ^[17].

Now, we introduce the $\varepsilon^{\beta}$ space and the order of growth of $f(z)\in{H(\mathbb{D})}$. Let $\beta\in(0, \infty)$ be a constant. Then $f(z)\in H(\mathbb{D})$ is said to belong to $\varepsilon^{\beta}$ space if and only if

for some constant $\alpha\in(0, \infty)$, which can be found in [7,p. 12]. Let $f(z)\in{H(\mathbb{D})}$. Then the order $\sigma_{M}(f)$ of $f(z)$ can be defined by

where $M(r, f) = \max\limits_{|z| = r}|f(z)|$. For a real number $x\geq 0$, the positive logarithm is defined as follows,

which can be found in [6,p. 3].

We say that $A$ and $B$ is comparable if there is a positive constant $C$ such that $\frac{B}{C}\leq A\leq CB$. The paper is organized as follows, the growth of solutions of differential equations is estimated in Section 2. The direct problem and inverse problem of differential equations are studied in Section 3 and Section 4 respectively. Finally, the higher order derivative of solutions of differential equations is characterized in Section 5.

2.   Growth of solutions

Auxiliary results. In order to prove Theorems 2.3 below. The following Lemma 2.1 plays an important role in dealing with the coefficients $A_{j}$ of (1.3).

Lemma 2.1. ^[17] Let $A(z)$ be an analytic function in $\mathbb{D}$, $1 < \alpha < \infty$ and $n\in \mathbb{N}$. Then the following quantities are comparable:

$\mathit{\rm{(i)}} \; ||A||_{H_{\alpha-1}^{\infty}}$,

$\mathit{\rm{(ii)}} \; ||A||_{\mathcal{B}^{\alpha}}+|A(0)|$,

$\mathit{\rm{(iii)}} \; \sup\limits_{z\in \mathbb{D}}|A^{(n)}(z)|(1-|z|^{2})^{n-1+\alpha}+\sum\limits_{j = 0}^{n-1}|A^{(j)}(0)|$.

In 2003, Chyzhykov-Gundersen-Heittokangas investigated the growth of solutions of (1.2) when the coefficient $A(z)\in{G_{p}}$, and obtained that every solution $f(z)$ of (1.3) satisfies $\sigma_{M}(f)\leq \frac{p}{k}-1$ for $p\geq k$, see [2,Theorem 2.2,p. 738] for more details. In 2010, Chyzhykov-Heittokangas-Rättyä studied the growth of solutions of (1.3) for $F(z) = 0$, and obtained more sharp results as Theorem 2.2 below.

Theorem 2.2. [3,Theorem 1.4,p. 147-148] Let $A_{j}(z)\in G_{p_{j}}$, $j = 0, 1, \ldots, k-1$, $F(z) = 0$ and $1\leq\alpha < \infty$, $p_{k} = 0$. If $f(z)$ is a solution of (1.3), then the following statements hold.

$\mathit{\rm{(i)}}$ $\sigma_{M}(f)\leq \left\{0, \max\limits_{0\leq j\leq k-1}\left\{\frac{p_{j}}{k-j}-1\right\}\right\}$ and

$\mathit{\rm{(ii)}}$ Suppose that $\min\limits_{1\leq j\leq k}\left\{\frac{p_{0}-p_{j}}{j}\right\}\geq 2$, then $\sigma_{M}(f)\leq\alpha$ if and only if $\max\limits_{0\leq j\leq k-1}\left\{\frac{p_{j}}{k-j}-1\right\}\leq\alpha$;

$\mathit{\rm{(iii)}}$ Suppose that $\min\limits_{1\leq j\leq k}\left\{\frac{p_{0}-p_{j}}{j}\right\}\geq 2$ holds. If $n\in\{0, 1, \ldots, k-1\}$ is the smallest index for which $\frac{p_{n}}{k-n} = \max\limits_{0\leq j\leq k-1}\left\{\frac{p_{j}}{k-j}-1\right\}$, then in every solution basis of (1.3) there are at least $k-n$ linearly independent solutions $f(z)$ such that $\sigma_{M}(f) = \max\limits_{0\leq j\leq k-1}\left\{\frac{p_{j}}{k-j}-1\right\}$.

Main result. Here we estimate the growth of solutions of (1.3) for the case of $F(z)\neq 0$, and prove the following result.

Theorem 2.3. Let $A_{j}(z)$ and $F(z)$ be analytic in $\mathbb{D}$, $j = 0, 1, \ldots, k-1$. Then the following statements hold.

$\mathit{\rm{(i)}}$ If $A_{j}(z)\in H_{q}^{\infty}$ and $F(z)\in H_{q}^{\infty}$, where $q\geq 1$, then every solution $f(z)$ of $(1.3)$ satisfies $\sigma_{M}(f)\leq q-1$;

$\mathit{\rm{(ii)}}$ If $A_{j}(z)\in H_{q}^{\infty}$ and $F(z)\in \varepsilon^{\beta}$, where $q\geq 1$ and $\beta > 0$, then every solution $f(z)$ of $(1.3)$ satisfies $\sigma_{M}(f)\leq \max\{q-1, \beta\}$.

Proof. (ⅰ) Since $A_{j}(z)\in H_{q}^{\infty}$ and $F(z)\in H_{q}^{\infty}$, then by Lemma 2.1, there exist constant $C_{1} > 0$ and $C_{2} > 0$ such that for any natural numbers $n$ and $j = 0, 1, \ldots, k-1$,

It follows from [10,Theorem 1 (a)] that there exist constant $C_{3} > 0$ and $C_{4} > 0$ such that for all $z = re^{i\theta}\in\mathbb{D}$,

Combining (2.1) and (2.2), there exist constant $C_{5} > 0$ and constant $C_{6} > 0$ such that

which implies that

(ⅱ) Since $F(z)\in \varepsilon^{\beta}$, then there exists a constant $\alpha > 0$ such that for all $z = re^{i\theta}\in \mathbb{D}$,

Combining (2.1), (2.2) and (2.3), there exists a constant $C_{7} > 0$ such that

which implies that

The following example shows that Theorem 2.3 is sharpness.

Examples. We consider the differential equation

(1) Let $A_{0}(z) = 0$, $A_{1}(z) = \frac{6}{(1-z)^{2}}$, $A_{2}(z) = \frac{-2}{(1-z)^{3}}+\frac{-6}{1-z}, \; F(z) = \frac{12}{(1-z)^{2}}+54$. Obviously, $f(z) = e^{\frac{1}{(1-z)^{2}}}+(z-1)^{3}$ is a solution of (2.4). It is easy to see that $A_{0}(z)$, $A_{1}(z)$, $A_{2}(z)$, $F(z)\in H_{3}^{\infty}$ and $\sigma_{M}(f) = 2$.

(2) Let $A_{0}(z) = \frac{54}{(1-z)^{3}}$, $A_{1}(z) = \frac{6}{(1-z)^{2}}$, $A_{2}(z) = \frac{-2}{(1-z)^{3}}+\frac{-6}{1-z}$ and $F(z) = \frac{54}{(1-z)^{3}}e^{\frac{1}{(1-z)^{2}}}+\frac{12}{(1-z)^{2}}$. It is easy to see that $A_{0}(z)$, $A_{1}(z)$, $A_{2}(z) \in H_{3}^{\infty}$, $F(z) \in \varepsilon^{2}$ and $f(z) = e^{\frac{1}{(1-z)^{2}}}+(z-1)^{3}$ is a solution of (2.4) with $\sigma_{M}(f) = 2$.

3.   Direct problem

Auxiliary results. Here we study the properties of function space of solution of (1.3) by limiting the condition of coefficients. In order to deal with the term $F(z)$ of (1.3), the Lemma 3.1 below is needed.

Lemma 3.1. [13,Lemma 2.5] Let $A(z)$ be analytic in $\mathbb{D}$. If $A^{(k)}(z)\in H^{p}(\frac{1}{k}\leq p\leq\infty$, $k\geq2)$, then $A(z)\in H^{\infty}$.

The following Lemma 3.2 can be proved by using the similar reason as in the proof of [11,Lemma 5.10 (Gronwall),p. 86], here we omit the details.

Lemma 3.2. Let $u$ and $v$ be nonnegative integrable functions in $[0, 1)$, and let $c > 0$ be a constant. If

then

Lemma 3.3. Let $g(\xi)$ be annalytic in $\mathbb{D}$. For any $z$, $z_{0}\in\mathbb{D}$ and $|\xi|\leq |z|$, set $h(z) = \int_{z_{0}}^{z}(z-\xi)^{k}g(\xi)d\xi$, where $k\in \mathbb{N}^{+}$. Then

holds for any $n\in \mathbb{N}$ and $0\leq n\leq k$.

Proof. If $k = 1$, then

Obviously, the conclusion holds.

If $k = 2$, then

And by the case $k = 1$, we get

Obviously, the conclusion holds.

If $k > 2$, then

Therefore,

Then we summarize that

By these equalities above, this proof is completed.

In 2014, the higher order non-homogenous linear differential equation

is studied by Li-Xiao in ^[13], where $A(z)$ and $F(z)\in H(\mathbb{D})$, which is improvement of previous results from [7,Theorem 4.3,p. 21].

Theorem 3.4. [13,Theorem 1.10] Let $A(z)$ and $F(z)$ be analytic in $\mathbb{D}$ satisfying

where $\alpha > 0$ and $\beta\geq0$ are finite constants, and $\frac{1}{k}\leq p\leq+\infty$. If $f(z)$ is a solution of (3.1), then the following statements hold.

$\mathit{\rm{(i)}}$ If $0\leq\beta < k$, then $f(z)\in H^{\infty}$;

$\mathit{\rm{(ii)}}$ If $\beta = k$, then $f(z)\in H_{\frac{\alpha}{(k-1)}}^{\infty}$;

$\mathit{\rm{(iii)}}$ If $k < \beta < \infty$, then $f(z)\in \varepsilon^{\beta-k}$.

Main results. Here we investigate the properties of solutions of (1.3) similarly to Theorem 3.4.

Theorem 3.5. Let $A_{j}(z)$ and $F(z)$ be analytic in $\mathbb{D}$ satisfying

where $\alpha > 0$ and $\beta\geq 0$ are finite constants, and $\frac{1}{k}\leq p\leq+\infty$, $j = 0, 1, \ldots, k-1$. If $f(z)$ is a solution of (1.3), then the following statements hold.

$\mathit{\rm{(i)}}$ If $0\leq\beta < k$, then $f(z)\in H^{\infty}$;

$\mathit{\rm{(ii)}}$ If $\beta = k$, then $f(z)\in H_{q}^{\infty}$, where $q$ only depending on $k$, $\alpha$ and $\beta$;

$\mathit{\rm{(iii)}}$ If $\beta > k$, then $f(z)\in \varepsilon^{\beta-k}$.

Proof. Set $g(z) = \frac{1}{(k-1)!}\int_{0}^{z}(z-\xi)^{k-1}F(\xi)d\xi, \; z\in\mathbb{D}$. By Lemma 3.3, we get

Combining Lemma 3.1 and $F(z)\in H^{p}$, we have

Hence, there exists a constant $C_{1} > 0$, such that

It follows from [10,Theorem 9,p. 152-153] that

where the constants $c_{n}\in\mathbb{C}$ depend on the initial values of $f(z)$, $f'(z)$, $\ldots$, $f^{(k-1)}(z)$, the constants $d_{j, n}\in\bf{Q}^{+}$, and the path of integration is a piecewise smooth curve in $\mathbb{D}$ joining $z_{0}$ and $z$. Let $z_{0} = 0$, $z = re^{i\theta}$ and $\xi = te^{i\theta}$. Combining this and (3.2), we obtain

where $C_{2}$ and $C_{3}$ are positive constants. It follows from this and Lemma 3.2 that

Next, we divide into three cases to estimate the derivative of the coefficients $A_{j}$.

Case 1: If $\beta = 0$, it is easy to see that

holds for any $q_{0}\in(0, 1)$. Then, there exists a constant $C'_{4} > 0$ such that for $n\in N$, and any $q_{0}\in(0, 1)$,

Combining (3.3) and (3.4), we get

for $\beta = 0$. Therefore, $f(z)\in H^{\infty}$ for $0 = \beta < k$.

Case 2: If $\max\{j:A_{j}\neq0\}\geq\beta > 0$, then there exist a constant $s\in\{j:A_{j}\neq0\}$ satisfying $s-1 < \beta\leq s$. By using the similar reason as in the proof of case 1, we get

and for any $q_{0}\in(0, 1)$

By Lemma 2.1, there exists a constant $C''_{4} > 0$ such that for every $n\in \mathbb{N}$,

and for any $q_{0}\in(0, 1)$,

Combining (3.3), (3.6) and (3.7), we get

for $\max\{j:A_{j}\neq0\}\geq\beta > 0$. Therefore, $f(z)\in H^{\infty}$ for $0 < \beta\leq\max\{j:A_{j}\neq0\} < k$.

Case 3: If $\max\{j:A_{j}\neq0\} < \beta$, then it follows from $|A_{j}(z)|\leq \frac{\alpha}{(1-|z|)^{\beta-j}}$ that

Then, by Lemma 2.1, there exists a constant $C'''_{4} > 0$ such that for every $n\in \mathbb{N}$ and $j = 0, 1, \ldots, k-1$,

It follows from (3.3) and (3.9) that there exists a constant $C_{5} > 0$ such that for all $z = re^{i\theta}\in \mathbb{D}$,

and then

which implies that

for $\max\{j:A_{j}\neq0\} < \beta < k$. Therefore, $f(z)\in H^{\infty}$ for $\max\{j:A_{j}\neq0\} < \beta < k$.

Therefore, by (3.5), (3.8) and (3.11), we get the conclusions (ⅰ) holds when $0\leq\beta < k$. It follows from (3.10) that the conclusions (ⅱ) holds when $\beta = k$; the conclusions (ⅲ) holds when $\beta > k$. This proof is completed.

The following examples show Theorem 3.5 is sharpness.

Examples. (1) Let

and

It follows from [7,Lemma 1.1.2,p. 8] that $F(z)\in H^{p}$ for any $p\in[\frac{1}{2}, \frac{2}{3})$, thus $A_{j}(z)$ and $F(z)$ satisfy the conditions of Theorem 3.5(i) with $\beta = \frac{5}{2}$, where $j = 0, 1, 2$. Obviously, $f(z) = (1-z)^{\frac{3}{2}}+\frac{5}{8}$ is a solution of (2.4) and $f(z)\in H^{\infty}$.

(2) Let

Here $A_{j}(z)$ and $F(z)$ satisfy the conditions of Theorem 3.5(ⅱ) for any $p\in[\frac{1}{2}, +\infty)$ and $\beta = 3$, $j = 0, 1, 2$. Obviously, $f(z) = \frac{1}{(1-z)^{2}}+\frac{1}{1-z}+(1-z)^{5}$ is a solution of (2.4) and $f(z)\in H_{2}^{\infty}$.

(3) Let

It is easy to see that $A_{0}(z), \; A_{1}(z)$, $A_{2}(z)$ and $F(z)$ satisfy the conditions of Theorem 3.5(ⅲ) with $\beta = 4$ and $\frac{1}{2}\leq p < 1$ ([4,exercise 1,p. 13] shows that $\frac{1}{1-z}\in H^{p}$ for any $p < 1$). And $f(z) = e^{\frac{1}{1-z}}+\frac{1}{24}(z-1)^{3}$ solves the equation (2.4) with $f(z)\in \varepsilon^{1}$.

It is well known that $f(z) = \sum a_{n}z^{n} \in H^{2}$ if and only if $\sum |a_{n}|^{2} < +\infty$ [4,p. 93]. Therefore, we obtained the following Corollary 3.6 by the Theorem 3.5.

Corollary 3.6. Let $A_{j}(z)$ and $F(z) = \sum a_{n}z^{n}$ be analytic in $\mathbb{D}$ satisfying

where $\alpha > 0$ and $\beta\geq 0$ are finite constants, and $j = 0, 1, \ldots, k-1$. If $f(z)$ is solution of (1.3), then the following statements hold.

$\mathit{\rm{(i)}}$ If $0\leq\beta < k$, then $f(z)\in H^{\infty}$;

$\mathit{\rm{(ii)}}$ If $\beta = k$, then $f(z)\in H_{q}^{\infty}$, where $q$ only depending on $k$, $\alpha$ and $\beta$;

$\mathit{\rm{(iii)}}$ If $\beta > k$, then $f(z)\in \varepsilon^{\beta-k}$.

Next, we get the following result by changing the condition of $F(z)$.

Theorem 3.7. Let $A_{j}(z)$ and $F(z)$ be analytic in $\mathbb{D}$ satisfying

where $\alpha_{1} > 0$, $\alpha_{1} > 0$, $\beta_{2}\geq0$, $\beta_{1}\geq 0$ are finite constants, $j = 0, 1, \; \ldots, \; k-1$. If $f(z)$ is solution of (1.3), then the following statements hold.

$\mathit{\rm{(i)}}$ If $0\leq \beta_{1} < k$ and $0\leq \beta_{2} < k$, then $f(z)\in H^{\infty}$;

$\mathit{\rm{(ii)}}$ If $0\leq \beta_{1} < k$ and $\beta_{2} = k$, then $f(z)\in H_{q}^{\infty}$ for any $q > 0$;

$\mathit{\rm{(iii)}}$ If $0\leq \beta_{1} < k$ and $\beta_{2} > k$, then $f(z)\in H_{\beta_{2}-k}^{\infty}$;

$\mathit{\rm{(iv)}}$ If $\beta_{1} = k$, then for all finite number $\beta_{2}$, there exist a constant $q > 0$ such that $f(z)\in H_{q}^{\infty}$;

$\mathit{\rm{(v)}}$ If $\beta_{1} > k$, then for any finite number $\beta_{2}$, $f(z)\in \varepsilon^{\beta_{1}-k}$.

Proof. By the conditions of Theorem 3.7 and using the similar way as in the proof of the case 3 of Theorem 3.5, then there exists a constant $C_{6} > 0$ such that for all non-negative integers $n$ and $j = 0, 1, \ldots, k-1$,

holds for the case $\max\{j:A_{j}\neq0\} < \beta_{1}$.

Combining (3.12), [10,Theorem 1 (a)] and the conditions of Theorem 3.7, there exist constant $C_{7} > 0$ and $C_{8} > 0$ such that

Let $h(z) = \exp\left(\int_{0}^{r}\frac{C_{8}}{(1-t)^{\beta_{1}-k+1}}dt\right)$. By using the similar reason as in the proof of Theorem 3.5, we get

Let $g(z) = \int_{0}^{r}\frac{C_{7}}{(1-t)^{\beta_{2}-k+1}}dt$ and by the condition of $F(z)$, we get

Therefore, by (3.13) and (3.14), we get

for $\beta_{1} = k$ and any finite $\beta_{2}$, and

for $\beta_{1} > k$ and any finite $\beta_{2}$. Therefore, these conclusions (iv)-(v) hold.

By the inequality (3.13), we get

for $\max\{j:A_{j}\neq0\} < \beta_{1} < k$. Now, we claim that if $|f(z)|\leq \log\frac{1}{1-r}$ for all $z = re^{i\theta}\in \mathbb{D}$, then $f(z)\in H^{\infty}_{q}$ for any $q > 0$. In fact,

Combining with (3.14) and (3.15), these conclusions (ⅰ)-(ⅲ) can be deduced when $\max\{j:A_{j}\neq0\} < \beta_{1} < k$.

For the case $0\leq \beta_{1}\leq \max\{j:A_{j}\neq0\}$, by (3.14), [10,Theorem 1] and using the similar way in the case 1 and case 2 of the proof of Theorem 3.5, there exists a constant $C^{*} > 0$ such that

which implies that these conclusions (ⅰ)-(ⅲ) hold for $0\leq \beta_{1}\leq \max\{j:A_{j}\neq0\}$. Then, these conclusions (ⅰ)-(ⅲ) can be deduced from these inequalities above. This proof is completed.

Next, some examples for Theorem 3.7 are given.

Examples. We consider the following second order differential equation

(1) Let $A_{0}(z) = \frac{1}{4(1-z)^{\frac{3}{2}}}$, $A_{1}(z) = \frac{1}{4(1-z)^{\frac{1}{2}}}$ and $F(z) = \frac{1}{4(1-z)^{\frac{3}{2}}}$ satisfying the conditions of Theorem 3.7 (ⅰ) with $\beta_{1} = \frac{3}{2}$ and $\beta_{2} = \frac{3}{2}$. Then $f(z) = \alpha e^{(1-z)^{\frac{1}{2}}}$+1 is a solution of (3.16) and $f(z)\in H^{\infty}$, where $\alpha\neq0$ is a finite constant.

(2) Let $A_{0}(z) = \frac{1}{1-z}$, $A_{1}(z) = \log(1-z)$ and $F(z) = \frac{2}{(1-z)^{2}}$ satisfying the conditions of Theorem 3.7 (ⅱ) with $\beta_{1}\in(1, 2)$ and $\beta_{2} = 2$. Then $f(z) = \log\frac{1}{1-z}$ is a solution of (3.16) and $f(z)\in H_{q}^{\infty}$ for any $q > 0$.

(3) Let $A_{0}(z) = \frac{1}{z-1}$, $A_{1}(z) = \frac{1}{3}$ and $F(z) = \frac{12}{(1-z)^{5}}$ satisfying the conditions of Theorem 3.7 (ⅲ) with $\beta_{1} = 1$ and $\beta_{2} = 5$. Then $f(z) = \frac{1}{(1-z)^{3}}$ is a solution of (3.16) and $f(z)\in H_{3}^{\infty}$.

(4) Let $A_{0}(z) = \frac{-5}{(1-z)^{2}}$, $A_{1}(z) = \frac{1}{2(z-1)}$ and $F(z) = \frac{-5}{(1-z)^{2}}$ satisfying the conditions of Theorem 3.7 (ⅳ) with $\beta_{1} = 2$ and $\beta_{2} = 2$. Then $f(z) = \frac{1}{(1-z)^{2}}+1$ is a solution of (3.16) and $f(z)\in H_{2}^{\infty}$.

(5) Let $A_{0}(z) = \frac{-6}{(1-z)^{4}}$, $A_{1}(z) = \frac{-2}{(1-z)^{3}}$ and $F(z) = \frac{-6}{(1-z)^{4}}$ satisfying the conditions of Theorem 3.7 (ⅴ) with $\beta_{1} = 4$ and $\beta_{2} = 4$. Then $f(z) = e^{\frac{1}{(1-z)^{2}}}+1$ is a solution of (3.16) and $f(z)\in \varepsilon^{2}$.

The following Theorem 3.8 is the generalization of [8,Theorem 3.3,p. 96], which they considered only the case of $F(z) = 0$ and $\beta = 1$.

Theorem 3.8. Let $0 < \delta < 1$. Suppose that $A_{j}(z)$ and $F(z)$ are analytic functions in $\mathbb{D}$ satisfying

and

where $\alpha_{1} > 0$, $\alpha > 0$, $\beta_{1}\geq0$ and $\beta\geq0$ are finite constants. If $f(z)$ is a solution of (1.3), then the following statements hold.

$\mathit{\rm{(i)}}$ If $0 < \beta < 1$, then $f(z)\in H_{\beta_{1}}^{\infty}$;

$\mathit{\rm{(ii)}}$ If $\beta = 1$, then $f(z)\in H_{q}^{\infty}$, where $q = \left\{\begin{aligned} & \beta_{1}+k^{2}\alpha^{\frac{1}{k}}, \quad 0 < \alpha < 1, \\ & \beta_{1}+k^{2}\alpha, \quad \alpha\geq 1, \end{aligned}\right.$;

$\mathit{\rm{(iii)}}$ If $\beta > 1$, then for any finite $\beta_{1}$, $f(z)\in \varepsilon^{\beta-1}$.

Proof. By [10,Theorem 2], there exists a constant $C_{9} > 0$ such that for all $\theta\in[0, 2\pi)$ and $0\leq s\leq r < 1$,

Combining the inequality above and the conditions of Theorem 3.8, we get

where $C_{10} > 0$ is a constant.

If $0 < \alpha < 1$, then

which means that $f(z)\in \left\{\begin{aligned} &H_{\beta_{1}+k^{2}\alpha^{\frac{1}{k}}}^{\infty}, \quad \beta = 1, \\ &H_{\beta_{1}}^{\infty}, \quad 0 < \beta < 1, \\ & \varepsilon^{\beta-1}, \quad \beta > 1. \end{aligned}\right.$

If $\alpha\geq1$, then

which means that $f(z)\in \left\{\begin{aligned} &H_{\beta_{1}+k^{2}\alpha}^{\infty}, \quad \beta = 1, \\ &H_{\beta_{1}}^{\infty}, \quad 0 < \beta < 1, \\ & \varepsilon^{\beta-1}, \quad \beta > 1. \end{aligned}\right.$ The proof is completed.

4.   Inverse problem

Auxiliary result. In order to prove Theorem 4.3, we need the following Lemma 4.1, which can be deduced from [2,Theorem 3.1].

Lemma 4.1. [2,Theorem 3.1] Let $k$ be integer satisfying $k\geq 0$, and let $\varepsilon > 0$ and $d\in{(0, 1)}$. If $f(z)$ is meromorphic in $\mathbb{D}$ such that $f$ does not vanish identically, then there exist a set $E\subset [0, 2\pi)$, which has linear measure zero, such that if $\theta\in [0, 2\pi)\backslash E$, then there is a constant $r_{0} = r(\theta)\in (0, 1)$ such that for all $\arg z = \theta$ and $|z|\in (r_{0}, 1)$,

where $s(r) = 1-d(1-r)$.

Main result. Now, we consider the following equation

where $A(z)$ and $F(z)$ are analytic functions in $\mathbb{D}$, in which the inverse problem is studied.

Theorem 4.2. Let $A(z)$ and $F(z)$ be analytic in $\mathbb{D}$, and $A(z)\in H^{\infty}_{q}$. If every solution $f(z)$ of (4.1) satisfies $f(z)\in \varepsilon^{\beta}$, then $F(z)\in \varepsilon^{\beta}$. Furthermore, if $q\leq k$, then every solution $f(z)$ of (4.1) satisfies $f(z)\in \varepsilon^{\beta}$ if and only if $F(z)\in \varepsilon^{\beta}$.

Proof. Suppose that $A(z)\in H^{\infty}_{q}$ and $f(z)$ is any solution of (4.1) satisfies $f(z)\in \varepsilon^{\beta}$. First, we prove that $f^{(k)}(z)\in \varepsilon^{\beta}$ holds for any integers $k\in N^{+}$. If $f^{(k)}(z)\not\in \varepsilon^{\beta}$, then there exist a constant $\beta' > \beta$ such that $|f^{(k)}(z)|\geq \exp\left(\frac{\alpha'}{(1-|z|)^{\beta'}}\right)$, which implies that $\sigma_{M}(f^{(k)})\geq \beta'$. It follows from [1,Proposition 1.2] that $\sigma_{M}(f) = \sigma_{M}(f^{(k)})\geq \beta'$. This contradicts with our hypothesis, and then $f^{(k)}(z)\in \varepsilon^{\beta}$ holds for any integers $k\in N^{+}$.

Next, we prove that $F(z)\in \varepsilon^{\beta}$. By (4.1) and the conditions of Theorem 4.2, we get

where $\alpha_{0} > 0$ is a constant depending only on $q$ and $\alpha$. This implies that $F(z)\in \varepsilon^{\beta}$.

Finally, we assume that $q\leq k$ and $F(z)\in \varepsilon^{\beta}$. By using the similar method as in the proof of Theorem 2.3, we get every solutions $f(z)$ of (4.1) satisfies $f(z)\in \varepsilon^{\beta}$. The proof is completed.

Theorem 4.3. Let $A(z)$ and $F(z)$ be analytic in $\mathbb{D}$. If every non-trivial solution $f(z)$ of (4.1) satisfies

where $\alpha > 0$ and $\beta > 0$ are finite constants. Then $A(z)\in H^{\infty}_{q}$, where $q$ is a positive constant depending only on $k$, $\beta$ and $p$.

Proof. Let $f(z)$ is a non-trivial solution of (4.1). By the conditions of Theorem 4.3, for any $\varepsilon > 0$, there exist a constant $r_{1}\in(0, 1)$ such that for all $r\in(r_{1}, 1)$,

By Lemma 4.1, for $d\in(0, 1)$, there exist a constant $r_{2}\in[0, 1)$ and a set $E$ of measure zero such that for all $r\in(r_{2}, 1)$ and $\theta \in [0, 2\pi)\backslash E$,

Combining (4.2), for all $r \in(\max\{r_{1}, r_{2}\}, 1)$, we get

By Eq (4.1), we have

It follows from the inequality above, (4.3) and the conditions of Theorem 4.3 that the conclusion holds.

5.   Higher derivative problem

For an analytic function $f(z)$, it is easy to see that $f'(z)$ and $f(z)$ do not necessarily belong to the same space. Let $f(z) = \frac{1}{1-z}$. Then $f(z)\in H_{1}^{\infty}$, and $f'(z) = \frac{1}{(1-z)^{2}}$. It is easy to get $f'(z)\in H_{2}^{\infty}$ and $f'(z)\not\in H_{1}^{\infty}$. Therefore, a natural problem is: what conditions on coefficients guaranteeing solution of differential equations and its derivative belong to the same space? Here we investigate this question to the linear differential equation

where $A(z)$ and $F(z)$ are analytic functions in $\mathbb{D}$. Firstly, we introduce some auxiliary results.

Auxiliary results. Here, an auxiliary results is given for the proof of our results.

Lemma 5.1. Let $f(z)$ be a solution of $(5.1)$. Then

where $C$ is a positive constant, and $m\in \mathbb{N}$ is a natural number satisfying $0\leq m\leq k-1$.

Proof. Let $f(z)$ be a solution of (5.1). By [10,Theorem 9], we get

Combining Lemma 3.3, we deduce that

where $0\leq m\leq k-1$. And then, we choose $z_{0} = 0$ and the path of integration to be the line segment $[0, z]$, for all $z = re^{i\theta}$ and $z\xi = se^{i\theta}\; (0\leq s\leq r)$,

where $C$ is a positive constant only depends on $f(0), f'(0), \ldots, f^{(k-1)}(0)$. The assertion follows from Lemma 3.2.

Main results. In fact, a simple estimation shows that $f(z)\in H^{\infty}_{q-1}$ when $f'(z)\in H^{\infty}_{q}\; (q > 1)$, but $f(z)\not\in H_{0}^{\infty} = H^{\infty}$ when $f'(z)\in H_{1}^{\infty}$, for example $f(z) = \log\frac{1}{1-r}$. So, it is very meaningful to study the properties of function space of derivative of solutions of differential equations.

Theorem 5.2. Let $A(z)$ and $F(z)$ be analytic in $\mathbb{D}$ satisfying

where $m$ is a positive integer satisfies $0 < m\leq k-2$ and $\frac{1}{k-m} \leq p\leq+\infty$, and $\alpha$ and $\beta$ are finite constants. If $f(z)$ is a solution of $(5.1)$, then the following statements hold.

$\mathit{\rm{(i)}}$ If $0\leq\beta < k-m$, then $f^{(m)}(z)\in H^{\infty}$;

$\mathit{\rm{(ii)}}$ If $\beta = k-m$, then $f^{(m)}(z)\in H_{q}^{\infty}$, where $q = \frac{\alpha}{(k-m-1)!}$;

$\mathit{\rm{(iii)}}$ If $\beta > k-m$, then $f^{(m)}(z)\in \varepsilon^{\beta-k+m}$.

Proof. By (5.3) and using similar method as in the proof of Theorem 3.5, we get

where $0 < C_{1} < +\infty$ and $C_{2} = \frac{\alpha}{(k-m-1)!(\beta-k+m)}$. It follows from the inequality above that these conclusion (ⅰ)-(ⅲ) hold.

The following Corollary 5.3 is also given by Theorem 5.2.

Corollary 5.3. Let $A(z)$ and $F(z) = \sum a_{n}z^{n}$ be analytic in $\mathbb{D}$ satisfying

where $\alpha$ and $\beta$ are finite constants. If $f(z)$ is a solution of $(5.1)$, then for $0\leq m \leq k-2$ the following statements hold.

$\mathit{\rm{(i)}}$ If $0\leq\beta < k-m$, then $f^{(m)}(z)\in H^{\infty}$;

$\mathit{\rm{(ii)}}$ If $\beta = k-m$, then $f^{(m)}(z)\in H_{\frac{\alpha}{(k-m-1)!}}^{\infty}$;

$\mathit{\rm{(iii)}}$ If $\beta > k-m$, then $f^{(m)}(z)\in \varepsilon^{\beta-k+m}$.

Theorem 5.4. Let $A(z)$ and $F(z)$ be analytic in $\mathbb{D}$ satisfying

where $\alpha_{1} > 0$, $\alpha_{2} > 0$, $\beta_{1}\geq 0$ and $\beta_{2}\geq 0$ are finite constants. If $f(z)$ is a solution of $(5.1)$, $m$ is a non-negative integer satisfies $m < k$, then the following statements hold.

$\mathit{\rm{(i)}}$ If $0\leq\beta_{1} < k-m$ and $0\leq\beta_{2} < k-m$, then $f^{(m)}(z)\in H^{\infty}$;

$\mathit{\rm{(ii)}}$ If $0\leq\beta_{1} < k-m$ and $\beta_{2} = k-m$, then $f^{(m)}(z)\in H_{q}^{\infty}$ for any $q > 0$;

$\mathit{\rm{(iii)}}$ If $0\leq\beta_{1} < k-m$ and $\beta_{2} > k-m$, then $f^{(m)}(z)\in H_{\beta_{2}-k+m}^{\infty}$;

$\mathit{\rm{(iv)}}$ If $\beta_{1} = k-m$ and $0\leq\beta_{2} < k-m$, then $f^{(m)}(z)\in H_{\frac{\alpha_{1}}{(k-m-1)!}}^{\infty}$;

$\mathit{\rm{(v)}}$ If $\beta_{1} = k-m$ and $\beta_{2} = k-m$, then $f^{(m)}(z)\in H_{\frac{\alpha_{1}}{(k-m-1)!}+1}^{\infty}$;

$\mathit{\rm{(vi)}}$ If $\beta_{1} = k-m$ and $\beta_{2} > k-m$, then $f^{(m)}(z)\in H_{\frac{\alpha_{1}}{(k-m-1)!}+\beta_{2}-k+m}^{\infty}$;

$\mathit{\rm{(vii)}}$ If $\beta_{1} > k-m$, then for any finite $\beta_{2}$, $f^{(m)}(z)\in \varepsilon^{\beta_{1}-k+m}$.

Proof. For all $z = re^{i\theta}\in\mathbb{D}$, set

and

By Lemma 5.1, there exists a constant $C_{3} > 0$ such that

From the conditions of Theorem 5.4, we get

and

where $C_{3} > 0$, $C_{4} = \frac{\alpha_{2}}{(k-m-1)!(\beta_{2}-k+m)}$, $C_{5} = \frac{\alpha_{1}}{(k-m-1)!(\beta_{1}-k+m)}$, $C_{6} = \frac{\alpha_{2}}{(k-m-1)!}$ and $C_{7} = \frac{\alpha_{1}}{(k-m-1)!}$ are constants. It follows from these inequalities that these conclusions (ⅰ)-(ⅶ) hold.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (Grant No. 11861023), the Foundation of Science and Technology of Guizhou Province of China (Grant No. [2015]2112) and the Foundation of Science and Technology project of Guizhou Province of China (Grant No. [2018]5769-05).

Conflict of interest

The authors declare no conflicts of interest in this paper.