On Common Borel Direction of Entire Function f and Its q-Difference Operator

Abstract

The common Borel direction of entire function f(z) and its q-difference operator is studied by using Nevanlinna theory. Some conditions of the existence of common Borel direction for entire function f(z) and its q-difference operator are obtained in this paper.

1 Introduction

In this paper, we assume that the reader is familiar with the basic results and standard notation of Nevanlinna theory of meromorphic functions, such as T(r, f), m(r, f), N(r, f), see [7, 16] for more details. Let f(z) be transcendental meromorphic function in $\mathbb {C}$ and let $\rho (f)$ and $\mu (f)$ denote the order and lower order of f(z), respectively, the definitions are showed as follows,

$$\begin{aligned} \rho (f)=\limsup _{r\rightarrow \infty }\frac{\log ^+T(r,f)}{\log r}, ~\mu (f)=\liminf _{r\rightarrow \infty }\frac{\log ^+T(r,f)}{\log r}, \end{aligned}$$

where $\log ^+x=\max \{\log x,0\}$. If f(z) is an entire function, then T(r, f) is replaced by $\log M(r,f)$, where $M(r,f)=\max \limits _{|z|=r}|f(z)|$.

For any $a\in \overline{\mathbb {C}}=\mathbb {C}\cup \{\infty \}$, let n(r, a, f) be the number of zeros of $f(z)-a$ (or poles of f(z) for $a=\infty$) in $\{z:|z|<r\}$ counting multiplicities. Denote

$$\begin{aligned} N(r,a,f)=\int _0^r\frac{n(t,a,f)}{t}dt, \\ m(r,a,f):={\left\{ \begin{array}{ll} \frac{1}{2\pi }\int _0^{2\pi }\log ^+|f(re^{i\theta })|d\theta ,~~&{}a=\infty ,\\ \frac{1}{2\pi }\int _0^{2\pi }\log ^+\frac{1}{|f(re^{i\theta })-a|}d\theta ,~~&{} a\in \mathbb {C}. \\ \end{array}\right. } \end{aligned}$$

Then Nevanlinna characteristic function of f(z) is defined as

$$\begin{aligned} T(r,a,f):=m(r,a,f)+N(r,a,f). \end{aligned}$$

If $a=\infty$, then

$$\begin{aligned} T(r,f)=m(r,f)+N(r,f). \end{aligned}$$

The following is an introduction to the definition of Borel direction. Before introducing the definition, it is necessary to define the relevant notions. For $\alpha , \beta \in [0,2\pi )$, $0<\beta -\alpha <2\pi$, $r>0$, define $\Omega (\alpha , \beta )=\{z: \alpha<\arg z<\beta \}$. For $a\in \mathbb {C}$, let $n\left( r,\Omega (\alpha ,\beta ), a,f\right)$ be the number of zeros of $f(z)-a$ or (poles of f(z) for $a=\infty$) in $\Omega (\alpha ,\beta )\cap \{z:|z|<r\}$ counting multiplicities.

Definition 1.1

[7, 16] Let f(z) be a meromorphic function with finite positive order $\rho$ and let $\varepsilon$ be any given positive constant, $\theta \in [0,2\pi )$. If

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log n(r,\Omega (\theta -\varepsilon ,\theta +\varepsilon ),a,f)}{\log r}=\rho \end{aligned}$$

(1)

holds for all $a\in \overline{\mathbb {C}}$ with at most two exceptions, then the ray $\arg z=\theta$ is a Borel direction of order $\rho$ of f(z), for short, we also say $\arg z=\theta$ is a Borel direction of f(z).

Borel direction is one of the important notion of Nevanlinna theory. The studying on Borel direction mainly has two aspects: (1) distribution of the Borel direction, they can be found in [5, 7, 11, 13, 18]. (2) existence of common Borel direction of meromorphic functions and its derivative, we refer [3, 10, 12, 14, 15, 17, 20] for the further details. In this article, we only focus on the common Borel direction problem. In 1928, Valiron proved the existence of Borel direction for meromorphic functions of finite positive order. It is well known that a function f(z) has the same order as its derivative $f^\prime (z)$. Therefore, Valiron raised the following question.

Question 1

[14] Let f(z) be a meromorphic function with finite positive order, whether f(z) and $f^\prime (z)$ have common Borel directions?

For the case of entire functions, many researchers have obtained partial results for Question 1, such as Valiron [15], Rauch [12], Chuang [3]. In [10], Milloux established the following theorem.

Theorem 1.1

[10] If f(z) is an entire function with finite positive order, then every Borel direction of its derivative $f^\prime (z)$ is also Borel direction of f(z).

It follows from Theorem 1.1 that Question 1 was solved for the case of entire functions with finite positive order. Moreover, the proof of Theorem 1.1 is quite complicated. Later, Yang and Zhang found some simplification and refinements in [17, 20], respectively.

If f(z) is a meromorphic function with infinite order, Sun completely solved Question 1 by type function and proximate order in [13]. For meromorphic function f(z) with finite positive order, the researchers have obtained partial results by adding some conditions, Yang and Zhang [19] generalized Milloux’s result by the following Theorem 1.2. However, Question 1 is until open for the case of meromorphic function with finite positive order.

Theorem 1.2

[19] Let f(z) be a meromorphic function with finite positive order and has a finite Borel exceptional value in the angle $\Omega (\theta -\varepsilon ,\theta +\varepsilon )$. If $\arg z=\theta \in [0,2\pi )$ is a Borel direction of f(z), then $\arg z=\theta$ is a Borel direction of $f^\prime (z)$.

Recently, the value distribution theory of complex difference operators is very interesting, see [2, 8] for more details. It comes in two main forms of difference operators, shift $\bigtriangleup _a$ and $q$-difference $D_q$, where $a\in \mathbb {C}\backslash \{0\}$, $q\in \mathbb {C}\backslash \{0, 1\}$, and

$$\begin{aligned} \bigtriangleup _af(z)=f(z+a)-f(z), ~~~~~D_qf(z)=f(qz)-f(z). \end{aligned}$$

Differential operators are very useful in numerical integration, numerical differentiation, and numerical solutions of differential equations. People always hope to transform non-stationary time series into stationary ones for mathematical processing, and the most commonly used transformation is differential transformation. For more properties of these difference operators are found in [2]. Note that the classical differential as the limit of family of $q$-difference operator $\widehat{D_q}f(z)=\frac{1}{qz-z}D_qf(z)$, see [9] for more details. The classical common Borel direction problem will be seen as limit of the deformation of the common Borel direction for f(z) and its the $q$-difference operator. In [9], Long et al studied the existence of the common Borel direction of the entire function f(z) and its q-difference operator. How about the existence of common Borel direction of entire function f(z) and $D_qf$ for the case of f(z) having faster growth and slower growth? Based on this purpose, we investigate this problem for the following two cases: one is hyper order $\rho _2(f)>0$, another is logarithmic order $\rho _{\log }(f)<\infty$. This article is divided into the following parts to describe our results. In Section 2, the case that the hyper order is infinite is considered. In Section 3, we consider the case of finite hyper order. The case of finite logarithmic order is considered in Section 4.

2 Entire function with infinite hyper order

We start from the concept of hyper order. For meromorphic function f(z), its hyper order, lower hyper order are given by

$$\begin{aligned} \rho _2(f)=\limsup _{r\rightarrow \infty }\frac{\log ^+\log ^+T(r,f)}{\log r}, \mu _2(f)=\liminf _{r\rightarrow \infty }\frac{\log ^+\log ^+T(r,f)}{\log r}, \end{aligned}$$

respectively. Next, we introduce the notions of hyper proximate order and type function.

Definition 2.1

[6] A positive function $\rho (r)$ defined on $[r_0,\infty )$ is a hyper proximate order of entire function f(z) in $\mathbb {C}$ provided the following conditions are satisfied.

• (i) $\rho (r)\rightarrow \rho _2(f)$ as $r\rightarrow \infty$;

• (ii) $\rho ^\prime (r)$ exists everywhere in $(0,\infty )$ except possibly in a countable set and

  $$\begin{aligned} \lim _{r\rightarrow \infty }r\rho ^\prime (r)\log r=0, \end{aligned}$$

  where $\rho ^\prime (r)$ denotes the derivative of $\rho (r)$;

• (iii) Let $U(r)=r^{\rho (r)}$,

  $$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log \log T(r,f)}{\log U(r)}=1. \end{aligned}$$

Definition 2.2

Suppose that $\rho (r)$ is hyper proximate order of entire function f(z) with infinite order in $\mathbb {C}$. A ray $\arg z=\theta \in [0,2\pi )$ from the origin is called a Borel direction of hyper proximate order $\rho (r)$ of f(z), if for any $\varepsilon >0$,

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log \log n(r,\Omega (\theta -\varepsilon ,\theta +\varepsilon ),a,f)}{\rho (r)\log r}=1 \end{aligned}$$

holds for all $a\in \overline{\mathbb {C}}$ with at most two exceptions.

Inspired by [9, Theorem 1], we obtained the following result.

Theorem 2.1

Let f(z) be an entire function satisfies $0<\mu _2(f)\le \rho _2(f)=\infty$, and let $\rho (r)$ be hyper proximate order of f(z). If f(z) satisfies

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{\log \log T(cr,f)}{\log \log T(r,f)}=1, \end{aligned}$$

where $c>1$, then for any $q\in \mathbb {R}\setminus \{0,1\}$, f(z) and $D_qf$ have at least one common Borel direction of hyper proximate order $\rho (r)$.

To prove the Theorem 2.1, we need the following Lemma 2.1 and Lemma 2.2, which are proved by using the similar reason as in the proof of [9, Lemma 1] and [9, Proposition 1] respectively.

Lemma 2.1

Let f(z) be a meromorphic function and $|q|\ne 1$. Then

$$\begin{aligned} \rho _2(D_qf)=\rho _2(f),~~\mu _2(D_qf)=\mu _2(f). \end{aligned}$$

Lemma 2.2

Let f(z) be an entire function of infinite order,and let $\rho (r)$ be hyper order proximate order of f(z). Denote $U(r)=r^{\rho (r)}$ is type function of f(z). If

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{\log \log T(cr,f)}{\log \log T(r,f)}=1, \end{aligned}$$

(2)

where $c>1$, then U(r) is also the type function of $D_qf$.

For the proof of Theorem 2.1, we also need the following Lemma 2.3.

Lemma 2.3

Let f(z) be an entire function with $\rho _2(f)=\infty$ and the type function $U(r)=r^{\rho (r)}$. If there exist $\theta \in [0,2\pi )$ and the sequence $\left\{ r_ne^{i\theta _n}\right\} _{n=1}^\infty$, such that

• (i) $\lim \limits _{n\rightarrow \infty }r_n=\infty ,~~~~~\lim \limits _{n\rightarrow \infty }\theta _n=\theta$;

• (ii) $\limsup \limits _{n\rightarrow \infty }\frac{\log \log \log |f(r_ne^{i\theta _n})|}{\rho (r_n)\log r_n}=1$.

Then, the ray $\arg z=\theta$ is a Borel direction of hyper proximate order $\rho (r)$ of f(z).

Proof

As the similar proof of [13, Theorem 2], without loss of generality, assume that $\theta =0$, then

$$\begin{aligned}{} & {} U(r)\sim \log \log n\left( r,\Omega (-\frac{\pi }{2i},\frac{\pi }{2i}),a,f\right) , i\rightarrow \infty , \\{} & {} \limsup _{r\rightarrow \infty }\frac{\log \log n\left( r,\Omega (-\frac{\pi }{2i},\frac{\pi }{2i}),a,f\right) }{\rho (r)\log r}=1. \end{aligned}$$

Hence, the ray $\arg z=\theta$ is a Borel direction of hyper proximate order $\rho (r)$ of f(z). $\square$

For the proof of Theorem 2.1, we need to do the following discussion. Define $\widetilde{\rho }(r): (0,\infty )\rightarrow (0,\infty )$,

$$\begin{aligned} \widetilde{\rho }(r)=\frac{\log \log T(r,f)}{\log r}. \end{aligned}$$

It’s obviously that $\widetilde{\rho }(r)$ is not the hyper proximate order. Next, we compare the relationship between $\widetilde{\rho }(r)$ and $\rho (r)$. If (2) holds, then

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{\widetilde{\rho }(cr)}{\widetilde{\rho }(r)}=1. \end{aligned}$$

It follows from this that

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{\log \log T(cr,f)}{\log \log T(r,f)}=1. \end{aligned}$$

Since $0<\mu _2(f)\le \rho _2(f)=\infty$, by the definition of $\widetilde{\rho }(r)$, we have $\liminf \limits _{r\rightarrow \infty }\widetilde{\rho }(r)>0$ and $\limsup \limits _{r\rightarrow \infty }\widetilde{\rho }(r)=\infty$. By

$$\begin{aligned} T(r,f)\le \log M(r,f)\le 3T(2r,f), \end{aligned}$$

(3)

then

$$\begin{aligned} 1\le \frac{\log \log \log M(r,f)}{\widetilde{\rho }(r)\log r}\le \frac{\log 2+\log r}{\log r} \frac{\widetilde{\rho }(2r)}{\widetilde{\rho }(r)}+\frac{\log \log 3}{\widetilde{\rho }(r)\log r}. \end{aligned}$$

Therefore

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{\log \log \log M(r,f)}{\widetilde{\rho }(r)\log r}=1. \end{aligned}$$

(4)

Proof of Theorem 2.1

Since

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log \log T(r,f)}{\rho (r)\log r}=1, \end{aligned}$$

then we can choose a sequence $\{r_n\}_{n=1}^\infty$ such that $\lim \limits _{n\rightarrow \infty }r_n=\infty$ and

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\widetilde{\rho }(r_n)}{\rho (r_n)}=1. \end{aligned}$$

(5)

There exists a sequence $\{\theta _n\}_{n=1}^\infty$ that converges to $\theta _0\in [0,2\pi )$ such that $M(r_n,f)=|f(r_ne^{i\theta _n})|$. Combining (4) and (5), then

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\log \log \log |f(r_ne^{i\theta _n})|}{\rho (r_n)\log r_n}=1. \end{aligned}$$

By Lemma 2.3, we know that $\arg z=\theta _0$ is a Borel direction of hyper proximate order $\rho (r)$ of f(z). We’re going to prove that the ray $\arg z=\theta _0$ is the Borel direction of hyper proximate order $\rho (r)$ of $D_qf$. We only need to prove that $D_qf$ satisfies condition $\mathrm (ii)$ of Lemma 2.3. On the one hand, since

$$\begin{aligned} |(D_qf)\left( |q|^{-1}r_ne^{i\theta _n}\right) |\ge M(r_n,f)-M(|q|^{-1}r_n,f)\ge \frac{1}{2}M(r_n,f), n\rightarrow \infty , \end{aligned}$$

then

$$\begin{aligned}&\liminf _{n\rightarrow \infty }\frac{\log \log \log |(D_qf)\left( |q|^{-1}r_ne^{i\theta _n}\right)| }{\widetilde{\rho }(|q|^{-1}r_n)\log (r_n)} \nonumber \\&\quad =\liminf _{n\rightarrow \infty }\frac{\log \log \log |(D_qf)\left( |q|^{-1}r_ne^{i\theta _n}\right)| }{\widetilde{\rho }(r_n)\log (r_n)} \nonumber \\&\quad \ge \liminf _{n\rightarrow \infty }\frac{\log \log \left( \log M(r_n,f)-\log 2\right) }{\widetilde{\rho }(r_n)\log (r_n)}=1. \end{aligned}$$

(6)

This implies that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{\log \log \log |(D_qf)\left( |q|^{-1}r_ne^{i\theta _n}\right)| }{\rho (r_n)\log (r_n)}\ge 1. \end{aligned}$$

It is obviously that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{\log \log \log |(D_qf)\left( |q|^{-1}r_ne^{i\theta _n}\right)| }{\rho (|q|^{-1}r_n)\log (|q|^{-1}r_n)}\ge 1. \end{aligned}$$

On the other hand,

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\frac{\log \log \log |(D_qf)\left( |q|^{-1}r_ne^{i\theta _n}\right)| }{{\rho }(|q|^{-1}r_n)\log (|q|^{-1}r_n)}\\&\quad \le \limsup _{n\rightarrow \infty }\frac{\log \log (\log M (r_n,f)+\log 2)}{{\rho }(|q|^{-1}r_n)\log (r_n)}\le \limsup _{n\rightarrow \infty }\frac{\log \log T(2r_n,f)}{{\rho }(|q|^{-1}r_n)\log (r_n)}\\&\quad =\limsup _{n\rightarrow \infty }\frac{\widetilde{\rho }(2r_n)\log (2r_n)}{{\rho }(|q|^{-1}r_n)\log (r_n)} =\limsup _{n\rightarrow \infty }\frac{\widetilde{\rho }(|q|^{-1}r_n)\log (2r_n)}{{\rho }(|q|^{-1}r_n)\log (r_n)}\\&\quad =\limsup _{n\rightarrow \infty }\frac{\widetilde{\rho }(|q|^{-1}r_n)\log (r_n)}{{\rho }(|q|^{-1}r_n)\log (r_n)}=1. \end{aligned}$$

It implies that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{\widetilde{\rho }(|q|^{-1}r_n)\log (r_n)}{{\rho }(|q|^{-1}r_n)\log (r_n)}=1. \end{aligned}$$

By Lemma 2.3, the ray $\arg z=\theta _0$ is also a Borel direction of hyper proximate order $\rho (r)$ of $D_qf$. This proof is completed. $\square$

3 Entire function with finite positive hyper order

In this section, we will consider the common Borel direction problem of f(z) and $D_qf$ for the case of finite positive hyper order. Firstly, let us review the definition of Borel direction of finite hyper order $\rho _2(f)$ of f(z). We only need to replace the numerator of (1) with $\log \log n(r,\Omega (\theta -\varepsilon ,\theta +\varepsilon ),a,f)$ and $\rho _2(f)$ in Definition 1.1, respectively.

In order to describe the following results, we need to introduce the growth of the hyper order of the function f(z) along with the ray $\arg z= \theta$.

Definition 3.1

Let f(z) be a transcendental entire function with finite positive hyper order $\rho _2(f)$, the hyper order growth of f(z) along with the ray $\arg z= \theta \in [0,2\pi )$ denote by $\rho _{2,\theta }(f)$, and is defined by

$$\begin{aligned} \rho _{2,\theta }(f)=\limsup _{r\rightarrow \infty }\frac{\log ^+\log ^+\log ^+|f(re^{i\theta })|}{\log r}. \end{aligned}$$

(7)

Obviously, $\rho _{2,\theta }(f)\le \rho _2(f)$ for any $\theta \in [0,2\pi )$. If $\rho _{2,\theta }(f)$ is not identically to the same constant in $[0,2\pi )$ almost everywhere, then f(z) is said to have non-trivial hyper order growth, on the contrary, f(z) has trivial hyper order growth. The following example illustrates that both situations are possible.

Example

• (i) Let $f_1(z)=\exp \{e^{z^2}+e^z\}$. Then $\rho _{2,\theta }(f_1)=0$ for $\theta \in [\frac{\pi }{2},\frac{3\pi }{4}]$ and $[-\frac{3\pi }{4},-\frac{\pi }{2}]$; $\rho _{2,\theta }(f_1)=1$ for $\theta \in [\frac{\pi }{4},\frac{\pi }{2}]$ and $[-\frac{\pi }{2},-\frac{\pi }{4}]$; $\rho _{2,\theta }(f_1)=2$ for $\theta \in [-\frac{\pi }{4},\frac{\pi }{4}]$ and $[-\frac{3\pi }{4},-\frac{5\pi }{4}]$. Then $f_1$ is said to have non-trivial hyper order growth.

• (ii) Let $f_2(z)=\exp \{\sin z\}$. Then $\rho _{2,\theta }(f_2)=1$ for $\theta \in (0,\pi )\cup (\pi ,2\pi )$; $\rho _{2,\theta }(f_2)=0$ for $\theta \in \{0,\pi \}$. Then $f_2$ is said to have trivial hyper order growth.

By using the Definition 3.1, we obtained the following result.

Theorem 3.1

Let f(z) be an entire function $\rho _2(f)$ satisfies $\rho _2(f)\in (\frac{1}{2},\infty )$ and f(z) has non-trivial hyper order growth. Then for any $q\in \mathbb {R}\backslash \{0,1\}$, f(z) and $D_qf$ have at least one common Borel direction of hyper order $\rho _2(f)$.

First, we need the following lemma in the proof of Theorem 3.1.

Lemma 3.1

Let f(z) be a transcendental entire function and $q\in \mathbb {R}\backslash \{0,1\}$. Then $\rho _{2,\theta }(f)=\rho _{2,\theta }(D_qf)$ for any $\theta \in [0,2\pi )$.

This proof is similar to [9, Lemma 2], and for the convenience of readers, its proof is provided.

Proof

Obviously, $\rho _{2,\theta }(f)\ge \rho _{2,\theta }(D_qf)$. Next, we prove $\rho _{2,\theta }(f)\le \rho _{2,\theta }(D_qf)$. Denote $k=\rho _{2,\theta }(D_qf)$. Then for any given $\varepsilon >0$, there exist two positive constants $C_1$ and $C_2$ such that

$$\begin{aligned} |f(qre^{i\theta })-f(re^{i\theta })|\le C_1+C_2\exp \left\{ \exp (r^{k+\varepsilon })\right\} \end{aligned}$$

holds for sufficient large $r>0$. Since

$$\begin{aligned} f(z)=\sum \limits _{n=0}^{m-1}\left( f(\frac{z}{q^n})-f(\frac{z}{q^{n+1}})\right) +f(\frac{z}{q^m}), \end{aligned}$$

then

$$\begin{aligned} |f(z)|\le \sum \limits _{n=0}^{m-1}\left| f(\frac{z}{q^n})-f(\frac{z}{q^{n+1}})\right| +|f(\frac{z}{q^m})|. \end{aligned}$$

Therefore, $\rho _{2,\theta }(f)\le \rho _{2,\theta }(D_qf)$, and then $\rho _{2,\theta }(f)=\rho _{2,\theta }(D_qf)$. Lemma 3.1 is completely proved. $\square$

In order to prove Theorem 3.1, the following lemma is also need. Before stating Lemma 3.2, some descriptions to $\rho _{2,\theta }(f)$ are needed. A ray $\arg z=\theta$ is called the ray of hyper order $\rho _2(f)$ of f(z) provided $\rho _{2,\theta }(f)=\rho _2(f)$. $\Omega (\alpha ,\beta )$ is called the ray angle domain of hyper order $\rho _2(f)$ of f(z) provided $\rho _{2,\theta }(f)=\rho _2(f)$ for every $\theta \in (\alpha ,\beta )$.

Lemma 3.2

Let f(z) be an entire function with the hyper order $\rho _2(f) \in (0 ,\infty)$. If $\arg z=\theta$ is a boundary of the ray angle domain of hyper order $\rho _2(f)$ of f(z), then $\arg z=\theta$ must be the Borel direction of hyper order $\rho _2(f)$ of f(z).

The following Lemmas 3.3–3.6 are needed for the proof of Lemma 3.2. The Lemma 3.3 is proved by using the similar reason in [4, Theorem 1].

Lemma 3.3

Assume that f(z) is analytic on $\Omega (-\frac{\pi }{2\beta }, \frac{\pi }{2\beta })$ and continuous on $\overline{\Omega }(-\frac{\pi }{2\beta }, \frac{\pi }{2\beta })$, where $\overline{\Omega }(-\frac{\pi }{2\beta }, \frac{\pi }{2\beta })$ is closure of $\Omega (-\frac{\pi }{2\beta }, \frac{\pi }{2\beta })$. For $\rho _0<\beta$,

$$\begin{aligned} |f(re^{\pm \frac{\pi }{2\beta }i})|\le \exp \{\exp (r^{\rho _0})\}, \end{aligned}$$

and

$$\begin{aligned} |f(re^{i\theta })|\le \exp \{\exp (r^\alpha )\} \end{aligned}$$

as $r\rightarrow \infty$ and $\theta \in (-\frac{\pi }{2\beta }, \frac{\pi }{2\beta })$. If $\alpha <\beta$, then the hyper order of f(z) does not exceed $\rho _0$ for $z\in \Omega (-\frac{\pi }{2\beta }, \frac{\pi }{2\beta })$.

Lemma 3.4

Let f(z) be an entire function with the hyper order $\rho _2(f)\in (0,\infty )$. Then for any $\theta \in (\theta _1, \theta _2)$, where $\theta _1$, $\theta _2$ are constants satisfies $\theta _2-\theta _1<\frac{\pi }{\rho _2(f)}$,

$$\begin{aligned} \rho _{2,\theta }(f)\le \max \{\rho _{2,\theta _1}(f),\rho _{2,\theta _2}(f)\}. \end{aligned}$$

Proof

Let $\rho _2^*=\max \{\rho _{2,\theta _1}(f),\rho _{2,\theta _2}(f)\}$, $\theta _2-\theta _1=\frac{\pi }{\alpha }$. Then $\rho _2^*\le \rho _2(f)<\alpha$, and for any $\varepsilon >0$,

$$\begin{aligned} |f(re^{i\theta _k})|\le \exp \left\{ \exp (r^{\rho _2^*+\varepsilon })\right\} \end{aligned}$$

hold on $\arg z=\theta _k, k=1,2$. Meanwhile,

$$\begin{aligned} |f(z)|\le \exp \left\{ \exp (r^{\rho _2(f)+\varepsilon })\right\} \end{aligned}$$

holds for $z\in D=\{z: \theta _1<\arg z<\theta _2\}$. By Lemma 3.3, for any $\theta \in (\theta _1, \theta _2)$,

$$\begin{aligned} \rho _{2,\theta }(f)\le \rho _2^*=\max \{\rho _{2,\theta _1}(f),\rho _{2,\theta _2}(f)\}. \end{aligned}$$

$\square$

Lemma 3.5

Let f(z) be an entire function with the hyper order $\rho _2(f)\in (0,\infty )$ and has non-trivial hyper order growth. Then f(z) has at least one ray angle domain of hyper order $\rho _2(f)$ and the measure of each such angle domain is not less than $\frac{\pi }{\rho _2(f)}$.

Proof

Firstly, it is obviously, there must be exists $\theta _0\in [0,2\pi )$, such that $\rho _{2,\theta _0}(f)=\rho _2(f)$.

Secondly, there must be exist ray angle domain $\Omega (\alpha ,\beta )$ of hyper order $\rho _2(f)$ of f(z) with $\beta -\alpha \ge \frac{\pi }{\rho _2(f)}$ and $\theta _0\in [\alpha ,\beta ]$. Otherwise, there are two non rays of hyper order $\rho _2(f)$ of f(z) $\arg z=\theta ^{\prime }$ and $\arg z=\theta ^{\prime \prime }$ on both sides of $\arg z=\theta _0$, satisfying $\alpha<\theta ^\prime<\theta ^{\prime \prime }<\beta$ and $\theta ^{\prime \prime }-\theta ^{\prime }<\frac{\pi }{\rho _2(f)}$, we have $\rho _{2,\theta ^\prime }(f)<\rho _2(f)$ and $\rho _{2,\theta ^{\prime \prime }}(f)<\rho _2(f)$. By Lemma 3.4, $\rho _2(\theta _0)\le \max \{\rho _{2,\theta ^\prime }(f), \rho _{2,\theta ^{\prime \prime }}(f)\}$, this is a contradiction.

Because of the arbitrariness of $\theta _0$, we know that $\arg z=\theta _0\in [0,2\pi )$ must be contained ray angle domain of hyper order $\rho _2(f)$ of f(z) or its boundary with the measure of the angle domain is not less than $\frac{\pi }{\rho _2(f)}$. Hence, the measure of any ray angle domain of hyper order $\rho _2(f)$ of f(z) is not less than $\frac{\pi }{\rho _2(f)}$. This conclusion is completely proved. $\square$

The following Lemma 3.6 is proved by using similar reason as in [5, Lemma 1].

Lemma 3.6

Let f(z) be an entire function with the hyper order $\rho _2(f)\in (0,\infty )$, and let $\rho _{2,\theta _0}(f)=\rho _0$. Then there exists a positive number $\alpha$, such that for sufficiently small positive number $\varepsilon$, when $|\theta -\theta _0|\le \alpha$,

$$\begin{aligned} |f(re^{i\theta })|\le C\exp \left\{ \exp \{r^{\rho _2(f)+\varepsilon }\sin (\theta -\theta _0)(\rho _2(f)+\varepsilon )+ r^{\rho _0+\varepsilon }\cos (\theta -\theta _0)(\rho _0+\varepsilon )\}\right\} , \end{aligned}$$

where C is a positive constant that only depends on $\theta _0$, $\rho _2(f)$ and $\varepsilon$.

Proof of Lemma 3.2

We divided into two cases to prove the conclusion.

Case 1. Suppose that $\arg z=\theta$ is not the common boundary of two ray angle domain of hyper order $\rho _2(f)$ and the clockwise direction of $\arg z=\theta$ represents ray angle domain of hyper order $\rho _2(f)$ of f(z). Assume that $\arg z=\theta$ is not Borel direction of hyper order $\rho _2(f)$ of f(z). We aim for a contradiction. Then there exist two distinct finite complex number $a_i(i=1,2)$ and the angle $\Omega (\theta -4\alpha , \theta +4\alpha )$, for sufficient large r and sufficient small $\alpha$,

$$\begin{aligned} \sum \limits _{i=1}^2 n(r, \Omega (\theta -4\alpha ,\theta +4\alpha ),a_i,f)<\exp \{r^\mu \}, 0<\mu <\rho _2(f). \end{aligned}$$

(8)

We consider the domain $G_r=\{z:\theta -4\alpha<\arg z<\theta +4\alpha , |z|<r\}$, take each $a_i(i=1,2)$ value point of f(z) as the center of the circle and $\frac{\alpha r}{4(N_r+1)}$ the radius and make exclusion circle $\Gamma _r$ in $G_r$, where $N_r=\sum \limits _{i=1}^2 n((1+4\alpha )r,\Omega (\theta -4\alpha , \theta +4\alpha ),a_i,f), i=1,2$.

For $\rho _0<\rho _2(f)$, We claim that,

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log ^+\log ^+\log ^+M(G^*,f)}{\log r}=\rho _0, \end{aligned}$$

(9)

where $M(G^*,f)=\max \limits _{z\in G^*}|f(z)|$, $G^*=\Omega (\theta +\alpha , \theta +3\alpha )$. Otherwise, there is always

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log ^+\log ^+\log ^+M(G^*,f)}{\log r}=\rho _2(f). \end{aligned}$$

By Lemma 3.4, $\max \{\rho _{2,\theta +\alpha }(f),\rho _{2,\theta +3\alpha }(f)\}=\rho _2(f)$. Hence, There exists a sequence of rays $\arg z=\theta _n$ in the counterclockwise direction from $\arg z=\theta$, such that $\theta _n\rightarrow \theta$ as $n\rightarrow \infty$. By Lemma 3.5, any the ray of hyper order must be located in ray angle domain and its measure not less than $\frac{\pi }{\rho _2(f)}$. Therefore, $\arg z=\theta$ must be the common boundary of two ray angle domains of hyper order $\rho _2(f)$ of f(z), which contradicts with the assumption.

Choosing a point $z_2$ on $\arg z=\theta -\alpha$ and $|z_2|=r_2$. Set $G_{r_2}=\{z: \theta -4\alpha<\arg z<\theta +4\alpha , |z|<r_2\}$. Due to the sum of the excluded circle diameters in $G_{r_2}$ is not greater than $\frac{\alpha r_2}{2}$. Therefore, we take $z_1\not \in \Gamma _r$ on $G_{r_2}\cap \{z: |z|=r_2\}$, draw a circle with $z_1$ as the center and $2\alpha r_2$ as the radius. By Poisson-Jensen formula, we obtain

$$\begin{aligned} \log |f(z_2)|\le \frac{2\alpha r_2+\alpha r_2}{2\alpha r_2-\alpha r_2}m(z_1, 2\alpha r_2, f)=3T(z_1, 2\alpha r_2, f), \end{aligned}$$

(10)

where $T(z_1, 2\alpha r_2,f)$ is the characteristic function of f(z) in circle domain which $z_1$ and $2\alpha r_2$ are the center and the radius. By [18, Lemma 2] and $f(z)\ne \infty$,

$$\begin{aligned} \begin{aligned}&T(z_1,2\alpha r_2,f)\le T\left( z_1,2\alpha r_2,\frac{f-a_1}{f-a_2}\right) +\log |f(z_1)-a_2|+C\\&\le C(\sum _{i=1}^2n(z_1,2\alpha r_2,a_i,f)+1)\Big \{\log \frac{4(N_{r_2}+1)(\sum \limits _{i=1}^2 n(z_1,2\alpha r_2,a_i,f)+1)}{\alpha r_2}\\&+\log ^+2\alpha r_2+\log ^+\frac{1}{3\alpha r_2}+\log 4\Big \}+\log \left| \frac{f(z_1)-a_1}{f(z_1)-a_2}\right| +\log |f(z_1)-a_2|+C, \end{aligned} \end{aligned}$$

(11)

where $n(z_1, 2\alpha r_2, a_i, f), i=1,2,$ is the number of zeros of $f(z)-a_i$ in circle domain which $z_1$ and $2\alpha r_2$ are the center and the radius and $N_{r_2}=\sum \limits _{i=1}^2 n((1+4\alpha )r_2,\Omega (\theta -4\alpha , \theta +4\alpha ),a_i,f)$.

Since,

$$\begin{aligned} \begin{aligned} n(z_1,2\alpha r_2,a_i,f)&<n((1+3\alpha )r,\Omega (\theta -4\alpha , \theta +4\alpha ),a_i,f)\\&\le \exp \Big \{(1+3\alpha )r_2\Big \}^\mu \le C \exp \{r_2^\mu \}, \end{aligned} \end{aligned}$$

as $r_2\rightarrow \infty$ and $N_{r_2}<Cr_2^\mu$, substitute equation (10), we have

$$\begin{aligned} \log |f(z_2)|<C\exp \{r_2^\mu \}\log r_2+6\log |f(z_1)|+C. \end{aligned}$$

By (9), we conclude

$$\begin{aligned} \log \log |f(z_2)|<Cr_2^\mu +\log \log r_2+Cr_2^{\rho _0+\alpha }, \end{aligned}$$

where $\rho _0+\alpha <\rho _2(f)$. This contradicts with $\arg z=\theta -\alpha$ is ray of hyper order $\rho _2(f)$ of f(z).

Case 2. Suppose that $\arg z=\theta$ is the common boundary of two angle domains of the hyper order $\rho _2(f)$ of f(z). Then $\rho _{2,\theta }(f)=\rho _0<\rho _2(f)$. Suppose that the contrary to the assertion that if $\arg z=\theta$ is not the Borel direction of hyper order $\rho _2(f)$ of f(z), then there exists two distinct complex number $a_i(i=1,2)$ and an angle domain $\Omega (\theta -5\alpha , \theta +5\alpha )$, such that for $0<\mu <\rho _2(f)$ and sufficiently large r,

$$\begin{aligned} \sum \limits _{i=1}^2 n\left( r, \Omega (\theta -5\alpha ,\theta +5\alpha ), a_i, f \right) <\exp \{r^\mu \}. \end{aligned}$$

Denote the angle domain $G_{r_3}=\{z: \theta -5\alpha<\arg z<\theta +5\alpha , |z|<(1+4\alpha )r\}$. In this angle domain $G_{r_3}$, we take the values $a_i(i=1,2)$ of f(z) at as the centers of circles and draw exclusion circles $\Gamma _r$ with a radius of $\frac{\alpha }{8(r^{2\rho _2(f)-1}(N_{r_3}+1))}$, where $N_{r_3}=\sum \limits _{i=1}^2n((1+4\alpha )r,\Omega (\theta -5\alpha ,\theta +5\alpha ),a_i,f), i=1,2$.

Let be $G_r^*=\{z:\theta -\alpha<\arg z<\theta +\alpha , |z|<(1+4\alpha )r\}$, take any point $z_2$ on $\arg z=\theta -2\alpha$ and $|z_2|=r_2$. Similarly, we can find $z_1\in G_r^*\cap \{z: |z|=r_2\}$ that satisfies $z_1\not \in \Gamma _{r}$. Take $z_1$ as the center and $\alpha r_2$ as the radius, by using the similar discussion for case 1,

$$\begin{aligned} \log |f(z_2)|<Cexp\{r_2^\mu \}\log r_2+10\log ^+|f(z_1)|+C. \end{aligned}$$

(12)

Due to the fact that the sum of the diameters of all the excluded circles does not exceed $\frac{\alpha }{4r_2^{2\rho _2(f)-1}}$, and $\rho _2(f)+\varepsilon<2\rho _2(f), \rho _0+\varepsilon <\rho _2(f)$, we can choose $z_1\in \Omega (\theta -\frac{\alpha }{2r_2^{2\rho _2(f)}},\theta +\frac{\alpha }{2r_2^{2\rho _2(f)}})$. By Lemma 3.6, for sufficiently large r,

$$\begin{aligned} \begin{aligned} \log \log |f(z_1)|&<r_2^{\rho _2(f)+\varepsilon }|\sin (\rho _2(f)+\varepsilon )\frac{\varepsilon }{2r_2^{2\rho _2(f)}}| +r_2^{\rho _0+\varepsilon }+C\\&<(\rho _2(f)+\varepsilon )\frac{r_2^{\rho _2(f)+\varepsilon -2\rho _2(f)}\alpha }{2}+r_2^{\rho _0+\varepsilon }+C\\&<r_2^{\rho _0+\varepsilon }+C. \end{aligned} \end{aligned}$$

Substitution equation (12), we have

$$\begin{aligned} \log \log |f(z_2)|<Cr_2^\mu +\log \log r_2+14r_2^{\rho _0+\varepsilon }+C. \end{aligned}$$

This is contradiction to $\arg z=\theta -2\alpha$ is ray of hyper order $\rho _2(f)$ of f(z). $\square$

Proof of Theorem 3.1

Without loss of generality, we assume that $|q|>1$. Since $\rho _2(f)\in (\frac{1}{2},\infty )$ and f(z) has non-trivial hyper order growth, then there exists an angle domain $\Omega (\alpha ,\beta )$ with $\beta -\alpha \ge \frac{\pi }{\rho _2(f)}$, such that $\rho _{2,\theta }(f)=\rho _2(f)$ for any $\theta \in (\alpha ,\beta )\in [0,2\pi )$. At the same time, there exists an opening angle strict included in $(0,2\pi )$. By Lemma 3.2, the boundary of the opening angle be the set of Borel directions of hyper order $\rho _2(f)$ of f(z), and by Lemma 3.1, it is also Borel direction of hyper order $\rho _2(D_q f)$ of $D_qf$. Then the proof of Theorem 3.1 is completed. $\square$

4 Entire function with finite logarithmic order

In this section, we will discuss common Borel direction problem of f(z) and D_qf(z) for the case of finite logarithmic order entire function f(z). Firstly, we recall the notion of logarithmic order and lower logarithmic order as follows,

$$\begin{aligned} \rho _{\log }(f)=\limsup _{r\rightarrow \infty }\frac{\log ^+T(r,f)}{\log \log r}, \mu _{\log }(f)=\liminf _{r\rightarrow \infty }\frac{\log ^+T(r,f)}{\log \log r}. \end{aligned}$$

In [1], Chern introduced the Borel direction of meromorphic function with finite logarithmic order.

Definition 4.1

[1] Let f(z) be meromorphic function in $\mathbb {C}$ with finite logarithmic order $\rho _{\log }(f)$ and let $\varepsilon$ be any given positive constant, $\theta \in [0,2\pi )$. If

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log n(r,\Omega (\theta -\varepsilon ,\theta +\varepsilon ),a,f)}{\log \log r}=\rho _{\log }(f)-1 \end{aligned}$$

holds with at most two possible exceptions, then the ray $\arg z=\theta$ is called a Borel direction of logarithmic order $\rho _{\log }(f)$ of f(z).

We may define the logarithmic order growth of f(z) along with the ray $\arg z=\theta$ by using similar way to Definition 3.1. We only need to replace the numerator of (7) with $\log ^+\log ^+|f(re^{i\theta })|$ and its denominator with $\log \log r$, denote $\rho _{\log ,\theta }(f)$. Obviously, $\rho _{\log ,\theta }(f)\le \rho _{\log }(f)$ for any $\theta \in [0,2\pi )$. If $\rho _{\log ,\theta }(f)$ is not identically to the same constant in $[0,2\pi )$ almost everywhere, then f(z) is said to have non-trivial logarithmic order growth. On the contrary, f(z) has trivial logarithmic order growth. The following Theorem 4.1 can be obtained by using the similar idea in Theorem 3.1.

Theorem 4.1

Let f(z) be an entire function with logarithmic order $\rho _{\log }(f)$ satisfies $\rho _{\log }(f)\in [1,\infty )$ and f(z) has non-trivial logarithmic order growth. Then for any $q\in \mathbb {R}\backslash \{0,1\}$, f(z) and $D_q(f)$ have at least one common Borel direction of logarithmic order $\rho _{\log }(f)-1$.

The proof of Theorem 4.1 is similar to Theorem 3.1 and will not be detailed in here.

Remark 1

Theorem 4.1 requires the logarithmic order of a function to be finite, does this conclusion hold for functions with infinite logarithmic order? The concept of proximate logarithmic order has been introduced in [1], but we cannot solve this problem by using the method of dealing with Theorem 2.1. Because the conclusion of Lemma 2.3 does not hold for this case. For example, the function $f(z)=e^z$, its logarithmic order $\rho _{\log }(f)$ is infinite and satisfies

$$\begin{aligned} \limsup \limits _{n\rightarrow \infty }\frac{\log \log |f(r_ne^{i\theta _n})|}{\rho _{\log }(r_n)\log \log r_n}=1, \end{aligned}$$

but $\arg z=0$ is not its Borel direction of infinite logarithmic order.