Fast growth and fixed points of solutions of higher-order linear differential equations 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 in the unit disc $D = \{z\in\mathbb{C}:|z| < 1\}$ (see ^[1,2,3,4]). As for the definition of the iterated order of meromorphic function, we know that for $r\in[0, 1)$, $\exp_{1}r = e^{r}$ and $\exp_{n+1}r = \exp(\exp_{n}r), \; n\in\mathbb{N}$, and for all $r$ sufficiently large in $(0, 1)$, $\log_{1}r = \log r$ and $\log_{n+1}r = \log(\log_{n}r), \; n\in\mathbb{N}$. Moreover, we denote by $\exp_{0}r = r, \; \log_{0}r = r, \; \exp_{-1}r = \log_{1}r,$ $\log_{-1}r = \exp_{1}r$. Then, let us recall the following definitions for $n\in\mathbb{N}$.

Definition 1 (^[5,6]). Let $f$ be a meromorphic function in $D$. Then the iterated $n$-order of $f$ is defined by

where $\log^{+}_{1}x = \log^{+}x = \max\{\log x, 0\}, \; \log^{+}_{n+1}x = \log^{+}(\log^{+}_{n}x).$ For $n = 1$, $\sigma_{1}(f) = \sigma(f)$.

If $f$ is analytic in $D$, then the iterated $n$-order is defined by

For $n = 1$, $\sigma_{M, 1}(f) = \sigma_{M}(f)$.

Remark 1 (^[5]). It follows by M. Tsuji ^[7] that if $f$ is an analytic function in $\mathbb{D}$, then

which is the best possible in the sense that there are analytic functions $g$ and $h$ such that $\sigma_{M, 1}(g) = \sigma_{1}(g)$ and $\sigma_{M, 1}(h) = \sigma_{1}(h)+1,$ see ^[8]. However, it follows by Proposition $2.2.2$ in ^[3] that $\sigma_{M, n}(f) = \sigma_{n}(f)$ for $n\geq 2$.

Definition 2 (^[5]). Let $f$ be a meromorphic function in $\mathbb{D}$. Then the iterated $n$-convergence exponent of the sequence of zeros in $\mathbb{D}$ of $f(z)$ is defined by

where $N(r, \frac{1}{f})$ is the integrated counting function of zeros of $f(z)$.

Similarly, the iterated $n$-convergence exponent of the sequence of distinct zeros in $\mathbb{D}$ of $f(z)$ is defined by

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

In ^[6], Heittokangas et al. investigated the fast growing solutions of the differential equations

where the coefficient of $f$ dominates other coefficients in the unit disc $D$. They proved the following result.

Theorem 1 (^[6]). Let $n\in\mathbb{N}$ and $\alpha\geq 0$. All solutions of $(1)$, where the coefficients $A_{0}(z), ..., A_{k-1}(z)$ are analytic in $D$, satisfy $\sigma_{M, n+1}(f)\leq\alpha$ if and only if $\sigma_{M, n}(A_{j})\leq\alpha$ for all $j = 0, ..., k-1$. Moreover, if $q\in\{0, ..., k-1\}$ is the largest index for which $\sigma_{M, n}(A_{q}) = \max\limits_{0\leq j\leq k-1}\{\sigma_{M, n}(A_{j})\}$, then there are at least $k-q$ linearly independent solutions $f$ of $(1.1)$ such that $\sigma_{M, n+1}(f) = \sigma_{M, n+1}(A_{q})$.

As a result of Theorem 1, by comparing the iterated n-order of coefficients, they obtained that if $q = 0$, i.e., $\sigma_{M, n}(A_{j}) < \sigma_{M, n}(A_{0})$, for all $j = 1, ..., k-1$, then all solutions $f\not\equiv 0$ of $(1.1)$ satisfy $\sigma_{n+1}(f) = \sigma_{M, n}(A_0)$ (see [6,Theorem 1.2]).

Heittokangas et al. in ^[6] and others also investigated the fast growth of solutions by comparing the iterated n-type of coefficients if $\sigma_{M, n}(A_{j})\leq\sigma_{M, n}(A_{0})$ for all $j = 1, ..., k-1$.

Cao and Yi obtained some results similar to Theorem 1 in ^[9], and in ^[5] Cao proved the following results in the cases of the modulus and characteristic function of the coefficient $A_0$ dominating, respectively, those of other coefficients.

Theorem 2 (^[5]). Let $H$ be a set of complex numbers satisfying $\overline{dens}_{D}\{|z|:z\in H\subseteq D\} > 0$, and let $A_{0}, \; A_{1}, ..., A_{k-1}$ be analytic functions in $D$ such that

and for some constants $0\leq\beta < \alpha$ we have, for all $\varepsilon > 0$ sufficiently small,

and

as $|z|\rightarrow 1^-$ for $z\in H$. Then every solution $f\not\equiv 0$ of $(1.1)$ satisfies $\sigma_{n}(f) = \infty$ and $\sigma_{n+1}(f) = \sigma_{M, n}(A_0)$.

Theorem 3 (^[5]). Let $H$ be a set of complex numbers satisfying $\overline{dens}_{D}\{|z|:z\in H\subseteq D\} > 0$, and let $A_{0}, \; A_{1}, ..., A_{k-1}$ be analytic functions in $D$ such that

and for some constants $0\leq\beta < \alpha$ we have, for all $\varepsilon > 0$ sufficiently small,

and

as $|z|\rightarrow 1^-$ for $z\in H$. Then every solution $f\not\equiv 0$ of $(1.1)$ satisfies $\sigma_{n}(f) = \infty$ and $\alpha_{M, n}\geq\sigma_{n+1}(f)\geq\sigma_{n}(A_0)$, where $\alpha_{M, n} = \max\{\sigma_{M, n}(A_j):j = 0, 1, ..., k-1\}$.

A result similar to Theorem 3 is given in Corollary 1 and 2.

In the case of the modulus of $A_{0}$ dominating those of other coefficients, Hamouda obtained extensive version and improved Theorem 2 in ^[10] as follows.

Theorem 4 (^[10]). Let $A_{0}(z), ..., A_{k-1}(z)$ be meromorphic functions in the unit disc $D$. If there exist a point $\omega_{0}$ on the boundary $\partial D$ of the unit disc and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that

for any $\mu > 0$, then every solution $f(z)\not\equiv 0$ of the differential Eq $(1.1)$ is of infinite order.

Theorem 5 (^[10]). Let $A_{0}(z), ..., A_{k-1}(z)$ be meromorphic functions in the unit disc $D$. If there exist $\omega_{0}\in \partial D$ and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that

where $n\geq 1$ is an integer, $(\exp_{1}(z) = \exp(z), \; \exp_{n+1}(z) = \exp\{\exp_{p}(z)\})$, and $\lambda > 0, \; \mu > 0$ are real constants, then every solution $f(z)\not\equiv 0$ of the differential Eq $(1.1)$ satisfies $\sigma_{n}(f) = \infty$, and furthermore $\sigma_{n+1}(f)\geq\mu$.

There arises a natural question: can we improve Hamouda's results further, and what is the extensive version releted to coefficient characteristic functions. The first aim of this paper is to investigate this problem and obtain the following results of the fast growth of solutions of $(1.1)$, where the modulus and characteristic function of $A_{0}$ dominates, respectively, those of other coefficients.

Theorem 6. Let $A_{0}(z), ..., A_{k-1}(z)$ be meromorphic functions in the unit disc $D$. Suppose there exist a point $\omega_{0}$ on the boundary $\partial D$ of the unit disc and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that for any $\mu > 0$,

If the Eq $(1.1)$ has a solution $f\not\equiv 0$, then f is of infinite order.

Theorem 7. Let $A_{0}(z), ..., A_{k-1}(z)$ be meromorphic functions in unit disc $D$. Suppose there exist $\omega_{0}\in \partial D$ and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that

where $n\geq 1$ is an integer, and $\lambda > 0, \; \mu > 0$ are real constants. If the Eq $(1.1)$ has a solution $f\not\equiv 0$, then f satisfies $\sigma_{n}(f) = \infty$ and $\sigma_{n+1}(f)\geq\mu$.

Remark 2. Obviously, Theorems 4 and 5 are direct results of Theorems 6 and 7.

Theorem 8. Let $A_{0}(z), ..., A_{k-1}(z)$ be analytic functions in the unit disc $D$. If there exist $\omega_{0}\in \partial D$ and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that for any $\mu > 0$,

then every solution $f\not\equiv 0$ of $(1.1)$ is of infinite order.

Theorem 9. Let $A_{0}(z), ..., A_{k-1}(z)$ be analytic functions in the unit disc $D$ such that

If there exist $\omega_{0}\in \partial D$ and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that

where $n\geq 1$ is an integer, and $\lambda > 0$ is a real constant, then every solution $f\not\equiv 0$ of $(1.1)$ satisfies $\sigma_{n}(f) = \infty$ and $\sigma_{n+1}(f) = \sigma_{M, n}(A_0)$.

Corollary 1. Let $A_{0}(z), ..., A_{k-1}(z)$ be analytic functions in the unit disc $D$ such that

If there exist $\omega_{0}\in \partial D$ and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that for some constants $0\leq\beta < \alpha$ and any given $\varepsilon\; (0 < \varepsilon < \sigma)$, we have

and

as $z\rightarrow \omega_{0}$ for $z\in\gamma$, then every solution $f\not\equiv 0$ of $(1.1)$ satisfies $\sigma_{n}(f) = \infty$ and $\sigma_{n+1}(f) = \sigma_{M, n}(A_0)$.

In fact, from the assumption of Corollary 1, for any given $\varepsilon\; (0 < \varepsilon < \sigma)$, taking $0 < \lambda < \alpha-\beta,$ we can easily obtain

Since $\varepsilon$ is arbitrary, we can substitute $\mu$ by $\sigma-\varepsilon$ in the assumption (1.5) and the proof of Theorem 9, and then easily obtain the result.

Corollary 2. Let $A_{0}(z), ..., A_{k-1}(z)$ be analytic functions in the unit disc $D$ such that

If there exist $\omega_{0}\in \partial D$ and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that for some constants $0\leq\beta < \alpha$ and any given $\varepsilon\; (0 < \varepsilon < \sigma)$, we have

and

as $z\rightarrow \omega_{0}$ for $z\in\gamma$, then every solution $f\not\equiv 0$ of $(1.1)$ satisfies $\sigma_{n}(f) = \infty$ and $\alpha_{M, n}\geq\sigma_{n+1}(f)\geq\sigma_{n}(A_0)$, where $\alpha_{M, n} = \max\{\sigma_{M, n}(A_j):j = 0, 1, ..., k-1\}$.

Remark 3. Corollary 1 improves Theorems 3 with an accurate value of $\sigma_{n+1}(f)$ instead of a range of it.

Example 1. Consider the following differential equation

where $H_{0}(z)$ and $H_{1}(z)$ are meromorphic functions in the unit disc $D$ and analytic at the point $\omega_{0} = 1.$ We choose the curve $\gamma$ to be the ray $\arg z = 0$ in $D$.

We note that if $\max\{\sigma_{2}(H_{0}), \; \sigma_{2}(H_{1})\} < 2,$ then the coefficients have the same 2-order and 2-type. It is easy to see by Theorem 6, every solution $f\not\equiv0$ of this equation satisfies $\sigma_{2}(f) = \infty$ and $\sigma_{3}(f)\geq2.$

We also have that if $1\leq|H_{1}(z)|\leq|H_{0}(z)|,$ then every solution $f\not\equiv 0$ of this equation satisfies $\sigma(f) = \sigma_{2}(f) = \infty$ and $\sigma_{3}(f)\geq2$. For example, if $H_{0}(z) = H_{1}(z) = \frac{1}{z},$ then the assumption $(1.3)$ holds since

Therefore, from Theorem 7, every solution $f\not\equiv 0$ of this equation satisfies $\sigma_{2}(f) = \infty$ and $\sigma_{3}(f)\geq2.$

If $H_{0}(z)$ and $H_{1}(z)$ above are analytic functions in $D$, then, from Theorems 8 and 9, the same results hold, and we can easily obtain $\sigma_{3}(f) = 2$ further.

In addition, Cao also investigated the fixed points of homogeneous linear differential equations in $D$ in ^[5].

Theorem 10 (^[5]). Under the hypothesis of one of Theorems 2 and 3, if $A_{1}(z)+zA_{0}(z)\not\equiv 0,$ then every solution $f\not\equiv 0$ of $(1.1)$ satisfies $\overline{\lambda}_{n+1}(f-z) = \sigma_{n+1}(f).$

The second aim of this paper is to investigate the fixed points of solutions of higher-order equation further. We obtain the following result.

Theorem 11. Let $A_{0}(z), ..., A_{k-1}(z)$ be finite iterated $n$-order analytic (or meromorphic) functions in the unit disc $D$. If all non-trivial solutions $f$ of $(1.1)$ satisfy $\sigma_{n}(f) = \infty$ and $\sigma_{n+1}(f) < \infty$, then $\overline{\lambda}_{n}(f-z) = \sigma_{n}(f) = \infty$ and $\overline{\lambda}_{n+1}(f-z) = \sigma_{n+1}(f).$

Remark 4. By removing the condition $A_{1}(z)+zA_{0}(z)\not\equiv 0,$ and as a general result, Theorems 11 improves Theorems 10.

Corollary 3. Assume that the assumptions of one of Theorems 2, 3, 9, Corollaries 1 and 2 hold. Then every solution $f\not\equiv 0$ of $(1.1)$ satisfies $\overline{\lambda}_{n+1}(f-z) = \sigma_{n+1}(f).$

2.   Preliminary Lemmas

Lemma 1 (^[8]). Let $k$ and $j$ be integers satisfying $k > j\geq0,$ and let $\varepsilon > 0$ and $d\in(0, 1)$. If $f$ is a meromorphic function in $D$ such that $f^{(j)}$ does not vanish identically, then

where $E\subset[0, 1)$ with finite logarithmic measure $\int_{E}\frac{dr}{1-r} < \infty$ and $s(|z|) = 1-d(1-|z|)$. Moreover, if $\sigma_{1}(f) < \infty$, then

while if $\sigma_{n}(f) < \infty$ for $n\geq 2$, then

Lemma 2. Let $f: D\to \mathbb{R}$ be an analytic or meromorphic function in the unit disc $D$. If there exist a point $\omega_{0}\in \partial D$ and a curve $\gamma\subset D$ tending to $\omega_{0}$ such that

then there exists a set $E\subset[0, 1)$ with infinite logarithmic measure $\int_{E}\frac{\rm{dr}}{1-r} = \infty$ such that for all $|z|\in E$, we have $f(z) < a.$

Proof of Lemma 2. Set

Then for any $\varepsilon = a-b > 0,$ there exists $\delta > 0$ such that for all $z\in\gamma$ and $0 < |z-\omega_0| < \delta,$ we have $f(z) < b+\varepsilon = a$. Let $g:z\to |z|, \; z\in\gamma$ and $E = \{|z|:z\in\gamma\cap D^{o}(\omega_{0}, \delta)\}.$ It is easy to see that $g$ is continuous and $E\subset[0, 1)$ is of infinite logarithmic measure. For all $|z|\in E,$ we have $z\in\gamma$ and $0 < |z-\omega_0| < \delta.$ Hence, for all $|z|\in E,$ we have $f(z) < a.$

Lemma 3 (^[4]). Let $f$ be a meromorphic function in the unit disc, and let $k\geq 1$ be an integer. Then

where $S(r, f) = O(\log^{+}T(r, f)+\log(\frac{1}{1-r})),$ possibly outside a set $E\subset[0, 1)$ with $\int_{E}\frac{\rm{dr}}{1-r} < +\infty.$ If $f$ is of finite order of growth, then

Lemma 4 (^[11]). Let $f$ be a meromorphic function in the unit disc $D$ for which $i(f) = p > 1$ and $\sigma_{p}(f) = \beta < +\infty,$ and let $k\geq 1$ be an integer. Then for any $\varepsilon > 0,$

holds for all $r$ outside a set $E\subset[0, 1)$ with $\int_{E}\frac{\rm{dr}}{1-r} < \infty.$

Lemma 5 (^[12]). Let $f$ be a solution of $(1.1)$ where the coefficients $A_{j}(z)\; (j = 0, ..., k-1)$ are analytic functions in the disc $D_{R} = \{z\in\mathbb{C}:|z| < R\}$, $0 < R\leq\infty,$ let $n_{c}\in\{1, ..., k\}$ be the number of nonzero coefficients $A_{j}(z), j = 0, ..., k-1,$ and let $\theta\in[0, 2\pi)$ and $\varepsilon > 0$. If $z_{\theta} = \nu e^{i\theta}\in D_{R}$ is such that $A_{j}(z_{\theta})\neq 0$ for some $j = 0, ..., k-1,$ then for all $\nu < r < R$,

where $C > 0$ is a constant satisfying

The next lemma follows by Lemma 5.

Lemma 6 (^[5,10]). Let $n\in\mathbb{N}$. If the coefficient $A_{0}(z), \; A_{1}(z), \cdots, A_{k-1}(z)$ are analytic in the unit disc $D$, then all solutions of $(1.1)$ satisfy $\sigma_{M, n+1}(f)\leq\max\{\sigma_{M, n}(A_{j}):j = 0, \cdots, k-1\}$.

Lemma 7 (^[9]). If $f$ and $g$ are meromorphic functions in the unit disc $D$, $n\in\mathbb{N}$, then we have

(i) $\sigma_{n}(f) = \sigma_{n}(1/f), \; \sigma_{n}(a\cdot f) = \sigma_{n}(f)\; (a\in\mathbb{C}-\{0\})$;

(ii) $\sigma_{n}(f) = \sigma_{n}(f')$;

(iii) $\max\{\sigma_{n}(f+g), \; \sigma_{n}(f\cdot g)\}\leq\max\{\sigma_{n}(f), \; \sigma_{n}(g)\}$;

(iv) if $\sigma_{n}(f) < \sigma_{n}(g)$, then $\sigma_{n}(f+g) = \sigma_{n}(g), \; \sigma_{n}(f\cdot g) = \sigma_{n}(g).$

Lemma 8 (^[11]). Let $A_{0}, \; A_{1}, ..., A_{k-1}$ and $F\; (\not\equiv 0)$ be finite iterated $p$-order analytic functions in the unit disc $D$. If $f$ is a solution with $\sigma_{p}(f) = \infty$ and $\sigma_{p+1}(f) = \rho < \infty$ of the equation

then $\overline{\lambda}_{p}(f) = \lambda_{p}(f) = \sigma_{p}(f) = \infty$ and $\overline{\lambda}_{p+1}(f) = \lambda_{p+1}(f) = \sigma_{p+1}(f) = \rho.$

Lemma 9 (^[13]). Let $A_{0}, A_{1}, ..., A_{k-1}$ and $F\not\equiv 0$ be meromorphic functions in the unit disc $D$ and let $f$ be a meromorphic solution of the differential equation

such that

Then $\overline{\lambda}_{p}(f) = \lambda_{p}(f) = \sigma_{p}(f)$.

3.   Proofs of Theorems

Proof of Theorem 6. Suppose that $f\not\equiv 0$ is a solution of $(1.1)$ with finite order $\sigma(f) = \sigma < \infty$. From Lemma 1, for a given $\varepsilon > 0$ there exists a set $E_1\subset[0, 1)$ with $\int_{E_1}\frac{\rm{dr}}{1-r} < \infty$, such that for all $z\in D$ satisfying $|z|\not\in E_1$, we have

From $(1.1)$ we can write

By $(3.1)$ and $(3.2)$, for all $z\in D$ satisfying $|z|\not\in E_1$, we have

By the assumption $(1.2)$ and Lemma 2, for any $\mu > 0$, there exists a set $E_2\subset[0, 1)$ with infinite logarithmic measure $\int_{E_2}\frac{\rm{dr}}{1-r} = \infty$ such that for all $|z|\in E_2$, we have

It yields that for any $\mu > 0$,

as $|z|\in E_{2}\setminus E_{1}$, where $E_{2}\setminus E_{1}$ is of infinite logarithmic measure. Obviously, $(3.4)$ contradicts $(3.3)$ in $\{z\in D:|z|\in E_2\setminus E_1\}$.

Proof of Theorem 7. Suppose that $f\not\equiv 0$ is a solution of $(1.1)$ with $\sigma_{n}(f) = \sigma_{n} < \infty$. From Lemma 1, for a given $\varepsilon > 0$ there exists a set $E_3\subset[0, 1)$ with $\int_{E_3}\frac{\rm{dr}}{1-r} < \infty$, such that for all $z\in D$ satisfying $|z|\not\in E_3$, if $n = 1,$ we have

and if $n\geq2,$ we have

Using $(3.5)$ and $(3.6)$ in $(3.2)$, we have

and

for all $z\in D$ satisfying $|z|\not\in E_3$. By the assumption $(1.3)$ and Lemma 2, there exists a set $E_4\subset[0, 1)$ with infinite logarithmic measure $\int_{E_4}\frac{\rm{dr}}{1-r} = \infty$ such that for all $|z|\in E_4$, we have

It yields that

Obviously, $(3.9)$ contradicts both $(3.7)$ and $(3.8)$ in $\{z\in D:|z|\in E_4\setminus E_3\}$. So, $\sigma_{n}(f) = \infty$. Now by Lemma 1 and since $\sigma_{n}(f) = \infty\; (n\geq 1)$, we have

where $E_5\subset[0, 1)$ with finite logarithmic measure and $s(|z|) = 1-d(1-|z|)\; (d\in(0, 1))$. By $(3.2)$ and $(3.10)$, for all $z\in D$ satisfying $|z|\not\in E_5$, we have

By $(3.9)$ and $(3.11)$, for all $z\in D$ satisfying $|z|\in E_4\setminus E_5$, we have

Set $s(|z|) = R.$ We have $1-|z| = \frac{1}{d}(1-R)$ and $\int_{E_5}\frac{dr}{1-r} < \infty$. So, $(3.12)$ becomes

Obviously, $d(E_4\setminus E_5)+1-d$ is of infinite logarithmic measure. Then by $(3.13)$, we get

Proof of Theorem 8. Suppose that $f\not\equiv 0$ is a solution of $(1.1)$ with finite order $\sigma(f) = \sigma < \infty$. From Lemma 3, we have

From $(1.1)$, we can write

It follows that

By the assumption $(1.4)$ and Lemma 2, for any $\mu > 0$, there exists a set $E_6\subset[0, 1)$ with infinite logarithmic measure such that for all $|z|\in E_6$, we have

It yields that for any $\mu > 0$,

Using $(3.14)$ and $(3.15)$, we get

It is easy to see $(3.17)$ contradicts $(3.16)$ in $\{z\in D:|z|\in E_6\}$. Therefore, $\sigma(f) = \infty$.

Proof of Theorem 9. Suppose that $f\not\equiv 0$ is a solution of $(1.1)$ with $\sigma_{n}(f) < \infty$. If $n = 1,$ we have $3.17$. If $n\geq2,$ from Lemma 4, for any $\varepsilon > 0$, we have

holds for all $r$ outside a set $E_{7}\subset[0, 1)$ with $\int_{E_{7}}\frac{\rm{dr}}{1-r} < \infty.$ By $(3.15)$ and $(3.18)$, we have

By the assumption $(1.5)$ and Lemma 2, there exists a set $E_8\subset[0, 1)$ with infinite logarithmic measure such that for all $|z|\in E_8$, we obtain

It yields that

Obviously, $(3.20)$ contradicts $(3.17)$ and $(3.19)$ in $\{z\in D:|z|\in E_8\setminus E_7\}$. So, $\sigma_{n}(f) = \infty$. Now by Lemma 3 and since $\sigma_{n}(f) = \infty\; (n\geq 1)$, we have

possibly outside a set $E_{9}\subset[0, 1)$ with $\int_{E_{9}}\frac{\rm{dr}}{1-r} < \infty.$ Using $(3.21)$ in $(3.15)$, we obtain

By $(3.20)$ and $(3.22)$, we have

Obviously, $E_8\setminus E_9$ is of infinite logarithmic measure. Then by $(3.23)$, we get

By Lemma 6, we obtain $\sigma_{n+1}(f)\leq\sigma_{M, n}(A_{0}) = \mu.$ Therefore, $\sigma_{n+1}(f) = \sigma_{M, n}(A_{0}) = \mu.$ We complete the proof.

Proof of Theorem 11. Suppose that $f\not\equiv 0$ is a solution of $(1.1)$. By the assumption, we have

Set $g(z) = f(z)-z, \; z\in D.$ Then by $(3.24)$, we get

Substituting $f = g+z$ into $(1.1)$, we get

Next we prove that $-A_{1}-zA_{0}\not\equiv 0$. Suppose that $-A_{1}-zA_{0}\equiv 0$. Hence Eq $(1.1)$ has a solution $f_{1}$ satisfying $f_{1} = -z$ and $\sigma_{n}(f_{1}) < \infty$. This contradicts $(3.24)$. Hence, $-A_{1}-zA_{0}\not\equiv 0$. By Lemma 7, we have

Hence, by $(3.25)$, $(3.26)$ and Lemma 8 or Lemma 9, we deduce that $\overline{\lambda}_{n}(g) = \sigma_{n}(g) = \infty, \; \overline{\lambda}_{n+1}(g) = \sigma_{n+1}(g)$. Therefore, we obtain

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11971042, 12071035 and 11561031).

Conflict of interest

The authors declare no conflicts of interest in this paper.