Meromorphic functions sharing three values with their derivatives in an angular domain

Abstract

In this paper, we investigate the problem of uniqueness transcendental meromorphic functions sharing three values with their derivatives in an arbitrary small angular domain including a singular direction. The obtained results extend the corresponding results from Gundersen, Mues–Steinmetz, Zheng, Li–Liu–Yi, and Chen.

1 Introduction and main result

Let $f: C\rightarrow \hat{C}= C\cup \{\infty \}$ be a meromorphic function, where C is the complex plane. We assume that the reader is familiar with the basic results and notations of Nevanlinna’s value distribution theory (see [6, 14, 15]) such as $T(r; f)$, $N(r,f)$, and $m(r,f)$. Meanwhile, the lower order μ and the order λ of a meromorphic function f are defined as follows:

$$\begin{aligned}& \mu :=\mu (f)=\liminf_{r\to \infty} \frac{\log T(r,f)}{\log r},\\& \lambda :=\lambda (f)=\limsup_{r\to \infty} \frac{\log T(r,f)}{\log r}. \end{aligned}$$

Let f and g be nonconstant meromorphic functions in the domain $D\subseteq C$. If $f-c$ and $g-c$ have the same zeros with the same multiplicities in D, then $c\in C\cup \{\infty \}$ is called a CM shared value in D of f and g. If $f-c$ and $g-c$ have the same zeros in D, then $c\in C\cup \{\infty \}$ is called an IM shared value in D of f and g. The zeros of $f-c$ imply the poles of f when $c=+\infty$.

In 1979, Gundersen [5] and Mues and Steinmetz [10] considered the uniqueness of a meromorphic function f and its derivative $f'$ and obtained the following result.

Theorem A

Let f be a nonconstant meromorphic function in C, and let $a_{j}$ ($j = 1, 2, 3$) be three distinct finite complex numbers. If f and $f'$ IM share $a_{j}$ ($j = 1, 2, 3$), then $f\equiv f'$.

Later on, Frank and Schwick [3] generalized this result and proved the following result.

Theorem B

Let f be a nonconstant meromorphic function, and let k be a positive integer. If there exist three distinct finite complex numbers a, b, and c such that f and $f^{(k)}$ IM share a, b, c, then $f\equiv f^{(k)}$.

In 2004, Zheng [16] first considered the uniqueness question of meromorphic functions with shared values in an angular domain and proved the following result (see [16, Theorem 3]).

Theorem C

Let f be a transcendental meromorphic function of finite lower order and such that $\delta =\delta (a, f^{(p)}) > 0$ for some $a\in C\cup \{\infty \}$ and an integer $p \geq 0$. Let the pairs of real numbers $\{\alpha _{j},\beta _{j}\}$ ($j=1,\ldots,q$) be such that

$$ -\pi \leq \alpha _{1}< \beta _{1}\leq \alpha _{2}< \beta _{2}\leq \cdots \leq \alpha _{q}< \beta _{q}\leq \pi , $$

with $\omega =\max \{\frac{\pi}{\beta _{j}-\alpha _{j}}:1\leq j\leq q \}$, and

$$ \sum_{j=1}^{q} (\alpha _{j+1} - \beta _{j}) < \frac {4}{\delta} \arcsin \sqrt{\delta \bigl(a, f^{(p)}\bigr)/2}, $$

where $\delta = \max \{\omega , \mu \}$. For a positive integer k, assume that f and $f^{(k)}$ IM share three distinct finite complex numbers $a_{j}$ ($j=1,2,3$) in $X=\bigcup_{l=1}^{q}\{z:\alpha _{j}\leq \arg z\leq \beta _{j}\}$. If $\omega < \lambda (f)$, then $f\equiv f^{(k)}$.

In 2015, Li, Liu, and Yi [8] observed that Theorem C is invalid for $q \geq 2$ and proved the following more general result, which extends Theorem C (see [8, p. 443]).

Theorem D

([8])

Let f be a transcendental meromorphic function of finite lower order $\mu (f)$ in C such that $\delta (a, f) > 0$ for some $a \in C$. Assume that $q \geq 2$ pairs of real numbers $\{\alpha _{j},\beta _{j}\}$ satisfy the conditions

$$ -\pi \leq \alpha _{1} < \beta _{1} \leq \alpha _{2} < \beta _{2} \leq \cdots \leq \alpha _{q} < \beta _{q} \leq \pi , $$

with $\omega =\max \{\frac{\pi}{(\beta _{j} -\alpha _{j})} : 1 \leq j \leq q\}$, and

$$ \sum_{j=1}^{q} (\alpha _{j+1} - \beta _{j}) < \frac {4}{\delta} \arcsin \sqrt{\delta (a, f)/2}, $$

(1.1)

where $\delta = \max \{\omega , \mu \}$. For a kth-order linear differential polynomial $L[f]$ in f with constant coefficients given by

$$ L[f] = b_{k} f^{(k)}+ b_{k-1} f^{(k-1)} + \cdots + b_{1} f', $$

(1.2)

where k is a positive integer, and $b_{k}\neq 0$, $b_{k-1}$, ⋅⋅⋅, $b_{1}$ are constants, assume that f and $L[f]$ IM share $a_{j}$ ($j = 1, 2, 3$) in

$$ X=\bigcup_{l=1}^{q}\{z:\alpha _{j}\leq \arg z\leq \beta _{j}\}, $$

where $a_{j}$ ($j = 1, 2, 3$) are three distinct finite complex numbers such that $a \neq a_{j}$ ($j = 1, 2, 3$). If $\lambda (f)\neq \omega$, then $f = L[f]$.

In 2019, Chen [1] proved the following result.

Theorem E

Let f be a nonconstant meromorphic function of lower order $\mu (f)>1/2$ in C, let $a_{j}$ ($j = 1, 2, 3$) be three distinct finite complex numbers, and let $L[f]$ be given by Theorem D. Then there exists an angular domain $D=\{z:\alpha \leq \arg z \leq \beta \}$, where $0\leq \beta -\alpha \leq 2\pi$, such that if f and $L[f]$ CM share $a_{j}$ ($j = 1, 2, 3$) in D, then $f = L[f]$.

Question 1.1

From Theorems C–E a natural question arises: whether we can get the corresponding results if the restriction of f on deficiency and lower order is removed or if restriction (1.1) for the width of the angular domain is removed. What is the relationship between these angular regions and the value distribution properties of f?

In theory of meromorphic functions, a function is uniquely determined by its value on a set with a accumulation point. It is natural to ask if we can prove similar results under the conditions

$$ \bar{E}_{D}(f, a_{j})=\bar{E}_{D} \bigl(f', a_{j}\bigr),\quad j=1,2,3, $$

for some typical set in C instead of a general angular domain in C, where $\bar{E}_{D}(a,f)=\{z:z\in D,f(z)=a\}$ (as a set in C).

In general, the answer of this question is negative. For $f(z)=e^{2 z}$, it is clear that $f(z)\ne f'(z)$, but $|f(z)|$ is bounded by 1 on the left-half plane D. Thus

$$ \bar{E}_{D}(f,n)=\bar{E}_{D}\bigl(f',n \bigr)=\emptyset \quad \text{for all } n>1. $$

This example shows us that if such an angular domain D exists, then it must be a region whose image under f is dense in C.

Based on the theory on singular direction for a meromorphic function (see [14]) and the research results on shared values of a meromorphic function (see [9, 11]), combining with Theorems D and E, we may conjecture that the angular domain of the singular direction may be right. In this paper, we investigate the above question and prove the following result, which extends Theorems D and E.

Theorem 1.1

Let f be a meromorphic function of finite order that satisfies $\lim_{r\to \infty}\sup \frac{T(r,f)}{(\log r)^{3}}=+\infty$, and let ε be an arbitrary small positive number. Then there exists a direction $\arg z=\theta _{0}$ ($0\leq \theta _{0}< 2\pi$) such that if f and $f'$ IM share three distinct finite complex numbers $a_{j}$ ($j=1,2,3$) in $A(\theta _{0},\varepsilon )=\{z: | \arg z-\theta _{0}|<\varepsilon \}$, then $f \equiv f'$.

Theorem 1.2

Let f be a meromorphic function of finite order that satisfies $\lim_{r\to \infty}\sup \frac{T(r,f)}{(\log r)^{3}}=+\infty$, let ε be an arbitrary small positive number, and let k be a positive integer, Then there exists a direction $\arg z=\theta _{0}$ ($0\leq \theta _{0}< 2\pi$) such that if f and $f^{(k)}$ CM share three distinct finite complex numbers $a_{j}$ ($j=1,2,3$) in $A(\theta _{0},\varepsilon )=\{z: | \arg z-\theta _{0}|<\varepsilon \}$, then $f \equiv f^{(k)}$.

To prove our main results, we introduce some notations about the Ahlfors–Shimizu character of a meromorphic function in C:

$$ T_{0}(r,f)= \int _{0}^{r}\frac{A(t)}{t}\,dt , \quad A(t)=\frac{1}{\pi} \int _{0}^{2 \pi} \int _{0}^{t}\biggl( \frac{ \vert f'(\rho e^{i\theta}) \vert }{1+ \vert f(\rho e^{i\theta}) \vert ^{2}} \biggr)^{2}\,d\rho \,d\theta . $$

Nevanlinna theory in an angular domain plays an important role in this paper,so we recall its fundamental notations. Let f be a meromorphic function in $D=\{z:\alpha \leq \arg z \leq \beta \}$, where $0\leq \beta -\alpha \leq 2\pi$. Nevanlinna [4] defined the following symbols:

$$\begin{aligned}& A_{\alpha ,\beta}(r,f)=\frac {\omega}{\pi} \int _{1}^{r}\biggl( \frac {1}{t^{\omega}}- \frac {t^{\omega}}{r^{2\omega}}\biggr) \bigl\{ \log ^{+} \bigl\vert f \bigl(te^{i \alpha}\bigr) \bigr\vert +\log ^{+} \bigl\vert f\bigl(te^{i\beta}\bigr) \bigr\vert \bigr\} \frac{dt}{t},\\& B_{\alpha ,\beta}(r,f)=\frac{2 \omega}{\pi r^{\omega}} \int _{\alpha}^{ \beta}\log ^{+} \bigl\vert f\bigl(re^{i\theta}\bigr) \bigr\vert \sin \omega (\theta -\alpha )\,d\theta ,\\& C_{\alpha ,\beta}(r,f)=2\sum_{1< \vert b_{m} \vert < r}\biggl( \frac {1}{ \vert b_{m} \vert ^{\omega}}-\frac{ \vert b_{m} \vert ^{\omega}}{r^{2\omega}}\biggr) \sin \omega (\theta _{m}-\alpha ),\\& S_{\alpha ,\beta}(r,f)=A_{\alpha ,\beta}(r,f)+B_{\alpha ,\beta}(r,f)+C_{ \alpha ,\beta}(r,f), \end{aligned}$$

where $\omega =\frac{\pi}{(\beta -\alpha )}$, and $b_{m}=|b_{m}|e^{i\theta _{m}}$ are the poles of f in D counting multiplicities.

2 Preliminaries

In this section, we prove some lemmas, which will be used in the proof of the main result.

Lemma 2.1

([2, 12])

Let $\mathcal {F}$ be a family of meromorphic functions such that for every function $f\in \mathcal {F}$, its zeros of multiplicity are at least k. If $\mathcal {F}$ is not a normal family at the origin 0, then for $0\leq \alpha \leq k$, there exist

• (a) a real number r ($0< r<1$),

• (b) a sequence of complex numbers $z_{n}\to 0$, $|z_{n}|< r$,

• (c) a sequence of functions ${f_{n}}\in \mathcal {F}$, and

• (d) a sequence of positive numbers $\rho _{n}\to 0$

such that

$$ g_{n}(z)={\rho _{n}}^{-\alpha}f_{n}(z_{n}+ \rho _{n}z) $$

converges locally uniformly with respect to spherical metric to a nonconstant meromorphic function $g(z)$ on C. Moreover, g is of order at most two.

For convenience, we use the following notation:

$$\begin{aligned} LD(r,f:c_{1},c_{2},c_{3}) =&c_{1} \Biggl[\sum_{i=1}^{4} m\biggl(r, \frac{f'}{f-a_{i}}\biggr)\Biggr] +c_{2}\Biggl[\sum _{i=1}^{4} m\biggl(r,\frac{f''}{f'-b_{i}}\biggr) \Biggr] \\ &{}+c_{3}\Biggl[ \sum_{i=1}^{4} m\biggl(r,\frac{f^{(k+1)}}{f^{(k)}-d_{i}}\biggr)\Biggr], \end{aligned}$$

where $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ ($i=1,2,3,4$) are finite complex numbers, and k is an integer such that $k\geq 2$.

Lemma 2.2

([11])

Let f be a meromorphic function in a domain $D=\{z:|z|< R\}$, let $a_{j}$ ($j=1,2,3$) be three distinct finite complex numbers, let t be a positive real number, and let $a\in C$. If

$$ \bar{E}_{D}(a_{j},f)=\bar{E}_{D} \bigl(ta_{j},f'\bigr) \quad \textit{for } j=1,2,3, $$

$a \neq a_{j}$, $f(0)\neq a_{j},\infty$ ($j=1, 2 ,3$), $f'(0)\neq 0,at$, $f''(0)\neq 0$, and $f'(0)\neq t f(0)$, then for $0< r< R$, we have

$$\begin{aligned} T(r,f) \leq& LD(r,f:2,3,0)+\log \frac{\prod_{i=1}^{3} \vert f(0)-a_{i} \vert ^{2} \vert f'(0)-ta_{i} \vert ^{3}}{ \vert tf(0)-f'(0) \vert ^{5} \vert f'(0) \vert ^{2}} \\ &{}+3\log \frac {1}{ \vert f''(0) \vert }+\biggl(\log ^{+}t+m\biggl(r, \frac{f''}{f'-ta}\biggr)+1\biggr)O(1), \end{aligned}$$

where $\bar{E}_{D}(a,f)=\{z:z\in D,f(z)=a\}$ (as a set in C), and $O(1)$ is a complex number depending only on a and $a_{i}$ ($i=1,2,3$).

Lemma 2.3

([13])

Let f, g be nonconstant meromorphic functions in the unit disc thath IM share distinct finite complex numbers $a_{1}$, $a_{2}$, $a_{3}$, and $a_{4}=\infty$. If $a \neq a_{j}$, $f(0)\neq a, a_{j}$ ($j=1, 2 ,3,4$), $f'(0)\neq 0, \infty$, and $f (0)\neq g (0)$, then

$$\begin{aligned} T(r,f) \leq & T(r,g)+\log \frac{\prod_{i=1}^{3} \vert f(0)-a_{i} \vert }{ \vert f'(0) \vert \vert f(0)-g(0) \vert } \\ &{}+O(1)\Biggl[m\biggl(r,\frac{f'}{f-a}\biggr)+\sum _{i=1}^{3} m\biggl(r,\frac{f'}{f-a_{i}} \biggr)+1\Biggr], \end{aligned}$$

where $O(1)$ is a complex number depending only on a and $a_{i}$ ($i=1,2,3$).

Lemma 2.4

Let f be a meromorphic function in a domain $D=\{z:|z|< R\}$, let $a_{1}$, $a_{2}$, $a_{3}$ be three distinct finite complex numbers, and let t be a positive real number. If

$$ E_{D}(a_{i},f)=E_{D}\bigl(ta_{i},f^{(k)} \bigr)\quad \textit{for } i=1,2,3, $$

$a \neq a_{j}$, $f(0)\neq a_{j},\infty$ ($j=1, 2 ,3$), $f^{(k)}(0)\neq 0,at$, $f^{(k+1)}(0)\neq 0$, and $f^{(k)}(0)\neq t f(0)$, then for $0< r< R$, we have

$$\begin{aligned} T(r,f) \leq & LD(r,f:1,0,1)+(k+1) \log \frac{\prod_{i=1}^{3} \vert f(0)-a_{i} \vert ^{2} \vert f^{(k)}(0)-ta_{i} \vert ^{3}}{ \vert tf(0)-f^{(k)}(0) \vert ^{5} \vert f^{(k)}(0) \vert ^{2}} \\ &{}+3(k+1)\log \frac {1}{ \vert f^{(k+1)}(0) \vert }+\biggl(\log ^{+}t+m\biggl(r, \frac{f^{(k+1)}}{f^{(k)}-ta}\biggr)+1\biggr)O(1), \end{aligned}$$

where $E_{D}(a,f)=\{z\in D: f(z)=a , \textit{counting multiplicity}\}$, and $O(1)$ is a complex number depending only on a and $a_{i}$ ($i=1,2,3$).

Proof

Since $E_{D}(a_{i},f)=E_{D}(ta_{i},f^{(k)})$ ($i=1,2,3$) with $t\ne 0$, from the assumptions we see that $f^{(k)}(z)\not \equiv tf(z)$. Therefore by the Nevanlinna basic theorem we have

$$\begin{aligned}& \sum_{j=1}^{3} N\biggl(r, \frac {1}{f-a_{j}}\biggr) \\& \quad \leq N\biggl(r,\frac {1}{tf-f^{(k)}}\biggr) \\& \quad \leq T\bigl(r,tf-f^{(k)}\bigr)+\log \frac {1}{ \vert tf(0)-f^{(k)}(0) \vert }= m \bigl(r,tf-f^{(k)}\bigr)+N\bigl(r,tf-f^{(k)}\bigr) \\& \quad \leq N\bigl(r,f^{(k)}\bigr)+m(r,f)+m\biggl(r,\frac{f^{(k)}}{f} \biggr)+\log ^{+}t+O(1) + \log \frac {1}{ \vert tf(0)-f^{(k)}(0) \vert } \\& \quad \leq T(r,f)+k\bar{N}(r,f)+m\biggl(r,\frac{f^{(k)}}{f}\biggr)+\log ^{+}t+O(1)+ \log \frac {1}{ \vert tf(0)-f^{(k)}(0) \vert }. \end{aligned}$$

Note that

$$\begin{aligned} \sum_{j=1}^{3} m\biggl(r, \frac {1}{f-a_{j}}\biggr) =& m\Biggl(r,\frac {1}{f^{(k)}} \sum _{j=1}^{3} \frac{f^{(k)}}{f-a_{j}}\Biggr)+O(1) \\ \leq & m\biggl(r,\frac {1}{f^{(k)}}\biggr)+ m\Biggl(r,\sum _{j=1}^{3} \frac{f^{(k)}}{f-a_{j}}\Biggr)+O(1). \end{aligned}$$

Therefore we have

$$\begin{aligned} \sum_{j=1}^{3} T\biggl(r, \frac {1}{f-a_{j}}\biggr) =&\sum_{j=1}^{3} m\biggl(r, \frac {1}{f-a_{j}}\biggr)+ \sum_{j=1}^{3} N\biggl(r,\frac {1}{f-a_{j}}\biggr) \\ \leq & T(r,f)+k\bar{N}(r,f)+m\biggl(r,\frac {1}{f^{(k)}} \biggr)+LD(r,f:1,0,0) \\ &{}+\log ^{+}t+\log \frac {1}{ \vert (tf-f^{(k)})(0) \vert }+O(1). \end{aligned}$$

Noting that $m(r,\frac {1}{f^{(k)}})\leq T(r,\frac {1}{f^{(k)}})= T(r,f^{(k)})+ \log \frac {1}{|f^{(k)}(0)|}$, by Nevanlinna’s first fundamental theorem we obtain

$$\begin{aligned} 2T(r,f) \leq & T\bigl(r,f^{(k)}\bigr)+k\bar{N}(r,f)+LD(r,f:1,0,0) \\ &{}+\log \frac{\prod_{i=1}^{3} \vert f(0)-a_{i} \vert }{ \vert (tf-f^{(k)})(0) \vert \vert f^{(k)}(0) \vert } + O(1)+ \log ^{+}t . \end{aligned}$$

Since $T(r,f^{(k)})\geq N(r,f^{(k)})=N(r,f)+ k\bar{N}(r,f)\geq (k+1) \bar{N}(r,f)$, implying that $\bar{N}(r,f)\leq T(r,f^{(k)})/(k+1)$, we have

$$\begin{aligned} 2T(r,f) \leq & \frac {2k+1}{k+1} T\bigl(r,f^{(k)} \bigr)+LD(r,f:1,0,0) \\ &{}+\log \frac{\prod_{i=1}^{3} \vert f(0)-a_{i} \vert }{ \vert (tf-f^{(k)})(0) \vert \vert f^{(k)}(0) \vert } + O(1)+ \log ^{+}t . \end{aligned}$$

(2.1)

On the other hand, note that from $\bar{E}_{D}(a_{i},f)=\bar{E}_{D}(ta_{i},f^{(k)})$, $i=1,2,3$, $\bar{E}_{D}(\infty ,f)= \bar{E}_{D}(\infty ,f^{(k)})$, and $f(0)\neq a_{j},\infty$ ($j=1, 2 ,3$) it follows that $f^{(k)}(0)\neq ta_{j},\infty$ ($j=1, 2 ,3$). By application of Lemma 2.3 to $f^{(k)}$ and tf we have

$$\begin{aligned} T\bigl(r,f^{(k)}\bigr) \leq& T(r,f)+LD(r,f:0,0,1))+\log \frac {\prod_{i=1}^{3} \vert f^{(k)}(0)-ta_{i} \vert }{ \vert tf(0)-f^{(k)}(0) \vert \vert f^{(k+1)}(0) \vert } \\ &{}+\biggl(\log ^{+}t+m\biggl(r,\frac{f^{(k+1)}}{f^{(k)}-ta}\biggr)+1 \biggr)O(1). \end{aligned}$$

(2.2)

Now substituting (2.2) into (2.1) we have

$$\begin{aligned} \frac {T(r,f)}{k+1} \leq& LD(r,f:1,0,1)+\log \frac{\prod_{i=1}^{3} \vert f(0)-a_{i} \vert \vert f^{(k)}(0)-ta_{i} \vert }{ \vert f^{(k+1)}(0) \vert \vert tf(0)-f^{(k)}(0) \vert ^{2} \vert f^{(k)}(0) \vert } \\ &{}+\biggl(\log ^{+}t+m\biggl(r,\frac{f^{(k+1)}}{f^{(k)}-ta}\biggr)+1 \biggr)O(1). \end{aligned}$$

Hence we have

$$\begin{aligned} T(r,f) \leq & LD(r,f:1,0,1)+(k+1)\log \frac{\prod_{i=1}^{3} \vert f(0)-a_{i} \vert ^{2} \vert f^{(k)}(0)-ta_{i} \vert ^{3}}{ \vert tf(0)-f^{(k)}(0) \vert ^{5} \vert f^{(k)}(0) \vert ^{2}} \\ &{}+ 3(k+1)\log \frac {1}{ \vert f^{(k+1)}(0) \vert }+\biggl(\log ^{+}t+m\biggl(r, \frac{f^{(k+1)}}{f^{(k)}-ta}\biggr)+1\biggr)O(1). \end{aligned}$$

This completes the proof of Lemma 2.4. □

Lemma 2.5

([7])

Let $f(z)$ be a meromorphic function in C. Let

$$ \beta _{p}(r)=\sup_{2\leq t\leq r} \biggl\{ \frac{T_{0}(t,f)}{(\log t)^{p}} \biggr\} , \qquad \varepsilon (r)=\biggl\{ \frac{1}{ \beta _{p}(r)}\biggr\} ^{\frac{1}{q}} $$

with $p\geq 2$ and $q\geq 3$. If $\lim_{r\to \infty}\beta _{p}(r)= \infty$, then there exist a sequence of positive numbers $\{r_{n}\}_{1}^{\infty}$ and a sequence of points $\{z_{n}\}_{1}^{\infty}$ in C such that $\lim_{n\to \infty} r_{n}=\lim_{n\to \infty}|z_{n}|=+ \infty$ and

$$ A\bigl(\varepsilon \bigl( \vert z_{n} \vert \bigr) \vert z_{n} \vert ,z_{n},f\bigr)\geq \frac{1}{64\pi ^{2}} \bigl\{ \beta _{p}(r_{n})\bigr\} ^{1-\frac{2}{q}} (\log r_{n})^{p-2} \quad (n=1,2, \ldots ), $$

(2.3)

where

$$ A(r,a,f)=\frac{1}{\pi} \int _{0}^{2\pi} \int _{0}^{r}\biggl( \frac{ \vert f'(a+\rho e^{i\theta}) \vert }{1+ \vert f(a+\rho e^{i\theta}) \vert ^{2}} \biggr)^{2}\,d\rho \,d\theta ,\quad \vert z_{n} \vert \leq r_{n}, $$

and

$$ T_{0}(r,f)= \int _{0}^{r}\frac{A(t)}{t}\,dt , \quad A(t)=\frac{1}{\pi} \int _{0}^{2 \pi} \int _{0}^{t}\biggl( \frac{ \vert f'(\rho e^{i\theta}) \vert }{1+ \vert f(\rho e^{i\theta}) \vert ^{2}} \biggr)^{2}\,d\rho \,d\theta . $$

Lemma 2.6

Let $f(z)$ be a meromorphic function satisfying the conditions of Lemma 2.5. Then there exist a direction $\arg z=\theta _{0}$ ($0\leq \theta _{0}< 2\pi$), a sequence of points $\{z_{n}\}$ ($|z_{n}|\to \infty$) with $\lim_{n\to \infty} \arg z_{n}=\theta _{0}$, and a sequence of real numbers $r_{n}$ with $\lim_{n\to \infty} r_{n}=+\infty$ such that (2.3) holds.

Proof

Set $z_{n}=|z_{n}|e^{i\theta _{n}}$ ($0\leq \theta _{n}<2\pi$) in Lemma 2.5. Since $\{\theta _{n}\}$ is a bounded sequence, there exists convergent subsequence, still denoted $\{\theta _{n}\}$. Set $\theta _{n}\to \theta _{0}$ ($n\to \infty$). Thus the lemma follows. □

We say that the direction $\arg z=\theta _{0}$ is an H direction of $f(z)$.

Lemma 2.7

([6, 17])

Let $f(z)$ be a meromorphic function in a domain $D=\{z:|z|< R\}$. If $f(0)\neq \infty$, then for $0< r< R$, we have

$$ \bigl\vert T(t,f)-T_{0}(t,f)-\log ^{+} \bigl\vert f(0) \bigr\vert \bigr\vert \leq \frac{1}{2} \log 2, $$

where $\log ^{+}|f(0)|$ is replaced by $\log |c(0)|$ when $f(0)=\infty$, $c(0)$ is the coefficient of the Laurent series of $f(z)$ at 0, and $T_{0}(t,f)$ is defined in (1.2).

Lemma 2.8

([9])

Let $f(z)$ be a nonconstant meromorphic function in the complex plane, and let $a_{1}$, $a_{2}$, $a_{3}$ be three distinct finite complex numbers. Assume that f and $f'$ IM share $a_{i}$ ($i = 1, 2, 3$) in $\Omega (\alpha ,\beta ) = \{z : \alpha < \arg z < \beta \}$ with $0 \leq \alpha <\beta < 2\pi$. Then one of the following two cases holds: (i) $f \equiv f'$, or (ii) $S_{\alpha ,\beta}(r, f) = Q(r,f)$, where $Q(r,f)$ is a quantity such that if $f(z)$ is of finite order, then $Q(r, f) =O(1)$ as $r\to \infty$, and if $f(z)$ is of infinite order, then $Q(r, f) =O(\log (rT(r,f)))$ as $r\notin E$ and $r\to \infty$, where E is a set of positive real numbers with finite linear measure.

Lemma 2.9

([4, 8])

Let f be a meromorphic function on $\overline{\Omega}(\alpha ,\beta )$. If $S_{\alpha ,\beta}(r, f)=O(1)$, then

$$ \log \bigl\vert f\bigl(re^{i\phi}\bigr) \bigr\vert =r^{\omega}c\sin \bigl(\omega (\phi -\alpha )\bigr)+o \bigl(r^{ \omega}\bigr) $$

uniformly for $\alpha \leq \phi \leq \beta$ as $r\notin F$ and $r\to \infty$, where c is a positive constant, $\omega =\frac{\pi}{\beta -\alpha}$, F is a set of finite logarithmic measure, and $\overline{\Omega}(\alpha ,\beta )=\{z : \alpha \leq \arg z \leq \beta \}$.

Lemma 2.10

([1])

Let f be a meromorphic function in C, let $a_{j}$ ($j=1,2,3$) be three distinct finite complex numbers, and let $L[f]$ be given by (1.2). Suppose that f and $L[f]$ CM share $a_{j}$ ($j=1,2,3$) in $D=\{z:\alpha \leq \arg z \leq \beta \}$, where $0<\beta -\alpha \leq 2\pi$. If $f\not \equiv L[f]$, then $S_{\alpha ,\beta}(r,f)=R(r,f)$, where $R(r,f)$ is a quantity such that if $f(z)$ is of finite order, then $R(r, f) =O(1)$ as $r\to \infty$, and if $f(z)$ is of infinite order, then $R(r, f) =O(\log (rT(r,f)))$ as $r\notin E$ and $r\to \infty$, where E is a set of positive real numbers with finite linear measure.

Lemma 2.11

([14])

Let $f(z)$ be a meromorphic function in disc $D(0,R)$ centered at 0 with radius R. If $f(0)\neq 0,\infty$, then for $0< r<\rho <R$, we have

$$ m\biggl(r,\frac{f^{(k)}}{f}\biggr)< c_{k}\biggl\{ 1+\log ^{+}\log ^{+} \biggl\vert \frac {1}{f(0)} \biggr\vert + \log ^{+}\frac {1}{r} +\log ^{+}\frac {1}{\rho -r}+\log ^{+}\rho + \log ^{+}T(\rho ,f)\biggr\} , $$

where k is a positive integer, and $c_{k}$ is a constant depending only on k.

Lemma 2.12

([14])

Let $T(r)$ be a continuous nondecreasing nonnegative function, and let $a(r)$ be a nonincreasing nonnegative function on $[r_{0},R]$ ($0< r_{0}< R<\infty$). If there exist constants b, c such that

$$ T(r)< a(r)+ b\log ^{+}\frac {1}{\rho -r}+c\log ^{+} T(\rho ) $$

for $r_{0}< r<\rho <R$, then

$$ T(r)< 2a(r)+B\log ^{+}\frac{2}{R-r}+C, $$

where B, C are two constants depending only on b, c.

The following inequalities in Lemmas 2.13 and 2.14 play an important role in the proof of the theorem.

Lemma 2.13

Let $f(z)$ be a meromorphic function with finite order $\lambda >0$, let $\arg z=\theta _{0}$ be a direction, let $\Gamma _{n}=\{z |z-z_{n}|<\varepsilon _{n}\}$ ($n=1,2,\ldots$) be a series of circles, where $z_{n}=|z_{n}|e^{i\theta _{n}}$, $\theta _{n}\to \theta _{0}$, $\lim_{n\to \infty}|z_{n}|=+\infty$, $\varepsilon _{n}=\epsilon _{n}|z_{n}|$, and $\lim_{n\to \infty}\epsilon _{n}=0$. Suppose that f and $f'$ IM share three distinct finite complex numbers $a_{j}$ ($j=1,2,3$) in $A(\theta _{0},\varepsilon )=\{z: | \arg z-\theta _{0}|<\varepsilon \}$. If $f\not \equiv f'$, then for every sufficiently large n ($n \geq n_{0}$),

$$ A(\varepsilon _{n},z_{n},f)\leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr). $$

(2.4)

Proof

Set $f_{n}(z)=f(z_{n}+\varepsilon _{n} z)$. We distinguish two cases.

Case 1. Assume that $f_{n}(z)$ is normal in $|z|\leq 1$, implying that

$$ \frac { \vert f'_{n}(z) \vert }{1+ \vert f_{n}(z) \vert ^{2}}= \frac{\varepsilon _{n}|f'(z_{n}+\varepsilon _{n} z)|}{1+ \vert f(z_{n}+\varepsilon _{n} z) \vert ^{2}}\leq M \quad (n=1,2,\ldots ) $$

in $|z|\leq 1$, where M is a positive number. Then we have

$$ A(\varepsilon _{n},z_{n},f)=\frac{1}{\pi} \int _{0}^{2\pi} \int _{0}^{ \varepsilon _{n}}\biggl( \frac{|f'(z_{n}+\rho e^{i\theta})|}{1+ \vert f(z_{n}+\rho e^{i\theta}) \vert ^{2}} \biggr)^{2} \rho \,d\rho \,d\theta \leq 2M^{2}. $$

So (2.4) holds.

Case 2. Assume that $f_{n}(z)$ is not normal in $|z|\leq 1$. By Lemma 2.1 there exist

• (1) a sequence of points $\{z'_{n}\}\subset \{|z|<1\}$;

• (2) a subsequence of $\{f_{n}(z)\}_{1}^{\infty}$ (without loss of generality, we still denote it by $\{f_{n}(z)\}$); and

• (3) positive numbers $\rho _{n}$ with $\rho _{n}\to 0$ ($n\to \infty$) such that

  $$ h_{n}(z)=f_{n}\bigl(z'_{n}+ \rho _{n}z\bigr)\to g(z) $$

  (2.5)

  in spherical metric uniformly on a compact subset of C as $n\to \infty$, where $g(z)$ is a nonconstant meromorphic function. Thus for any positive integer k, we have

  $$ h_{n}^{(k)}(\xi )={\rho _{n}}^{k} f_{n}^{(k)}\bigl(z'_{n}+\rho _{n}\xi \bigr) \to g^{(k)}(\xi ). $$

We claim that $g''(\xi )\not \equiv 0$. Otherwise, $g(z)=cz+d$ ($c ,d\in \mathbf{C}$ and $c\neq 0$). We can choose $\xi _{0}$ with $g(\xi _{0})=a_{1}$. By Hurwitz’s theorem there exists a sequence $\xi _{n}\to \xi _{0}$ such that

$$ h_{n}(\xi _{n})=f_{n} \bigl(z'_{n}+\rho _{n}\xi _{n}\bigr)=g(\xi _{0})=a_{1}. $$

Notice that f and $f'$ IM share $a_{1}$ in $\{z: |\arg z-\theta _{0}|<\varepsilon \}$ and $s\neq \infty$, so we have

$$\begin{aligned} c&=g'(\xi _{0})=\lim_{n\to \infty} h'_{n}(\xi _{n})=\lim _{n\to \infty}\rho _{n}\varepsilon _{n}f' \bigl(z_{n}+\varepsilon _{n} \bigl(z'_{n}+ \rho _{n}\xi _{n}\bigr)\bigr)\\ &=\lim_{n\to \infty}\rho _{n}\varepsilon _{n}f\bigl(z_{n}+\varepsilon _{n} \bigl(z'_{n}+ \rho _{n}\xi _{n}\bigr)\bigr)=\lim_{n\to \infty}\rho _{n}\varepsilon _{n} a_{1}, \end{aligned}$$

and thus

$$ \lim_{n\to \infty}\rho _{n}\varepsilon _{n}=\frac {c}{a_{1}}. $$

Likewise, we get

$$ \lim_{n\to \infty}\rho _{n}\varepsilon _{n}=\frac {c}{a_{2}}, $$

which gives a contradiction.

For a sequence of positive numbers $\rho _{n}\varepsilon _{n}$, it is easy to snow that there exists a subsequence, still denoted by $\rho _{n}\varepsilon _{n}$, such that $\lim_{n\to \infty} \rho _{n}\varepsilon _{n}=a_{0}$, where $a_{0}\in [0,+\infty )\cup \{+\infty \}$. Now we consider two cases: $a_{0}=0$ or +∞, and $0< a_{0}<+\infty$.

Case 1. Assume that $\lim_{n\to \infty}\rho _{n}\varepsilon _{n}=0$ or ∞.

We choose $\xi _{0}\in C$ such that

$$ g(\xi _{0})\ne 0,a_{1},a_{2},a_{3}, \infty , \qquad g'(\xi _{0})\ne 0,\infty ,\qquad g''(\xi _{0})\ne 0,\infty . $$

Let

$$ p_{n}(z)=f_{n}\bigl(z'_{n}+ \rho _{n}\xi _{0}+z\bigr) $$

for arbitrary small $\varepsilon >0$. In view of

$$ \overline{E}_{A(\theta _{0},\varepsilon )} (a_{j}, f)=\overline{E}_{A( \theta _{0},\varepsilon )} \bigl(a_{j}, f'\bigr),\quad j=1,2,3, $$

and $\lim_{n\to \infty}\epsilon _{n}=0$, for sufficiently large n, we have

$$ \Gamma _{n}=\bigl\{ z \vert z-z_{n} \vert < \epsilon _{n} \vert z_{n} \vert ,z_{n}= \vert z_{n} \vert e^{i \theta _{0}}\bigr\} \subseteq A(\theta _{0},\varepsilon /2). $$

Therefore for every sufficiently large n ($n\geq n_{0}$), on $|z|\leq 4$, we have

$$ \bar{E}_{D}\bigl(a_{i},p_{n}(z)\bigr)= \bar{E}_{D}\bigl(\varepsilon _{n} a_{i},p'_{n}(z) \bigr)\quad (i=1,2,3). $$

Note that

$$\begin{aligned}& p_{n}(0)=f_{n}\bigl(z'_{n}+ \rho _{n}\xi _{0}\bigr)=h_{n}(\xi _{0})\to g(\xi _{0}) \ne a_{1},a_{2},a_{3}, \infty ,\\& p'_{n}(0)= f'_{n} \bigl(z'_{n}+\rho _{n}\xi _{0}\bigr)= \frac{h'_{n}(\xi _{0})}{ \rho _{n}},\qquad h'_{n}( \xi _{0})\to g'(\xi _{0}),\\& p''_{n}(0)=f''_{n} \bigl(z'_{n}+\rho _{n}\xi _{0}\bigr)= \frac{h''_{n}(\xi _{0})}{ \rho _{n}^{2}},\qquad h''_{n}( \xi _{0})\to g''(\xi _{0}),\\& \varepsilon _{n}p_{n}(0)-p'_{n}(0)= \frac{\varepsilon _{n}\rho _{n}h_{n}(\xi _{0})-h'_{n}(\xi _{0})}{ \rho _{n}} . \end{aligned}$$

Thus we have

$$\begin{aligned} &\log \frac{\prod_{i=1}^{3} \vert p_{n}(0)-a_{i} \vert ^{2} \vert p'_{n}(0)-\varepsilon _{n}a_{i} \vert ^{3}}{ \vert \varepsilon _{n} p_{n}(0)-p'_{n}(0) \vert ^{5} \vert p'_{n}(0) \vert ^{2}}+3\log \frac{1}{ \vert p''_{n}(0) \vert } \\ &\quad =\log \frac{\prod_{i=1}^{3} \vert p_{n}(0)-a_{i} \vert ^{2} \vert p'_{n}(0)-\varepsilon _{n} a_{i} \vert ^{3}}{ \vert \varepsilon _{n} p_{n}(0)-p'_{n}(0) \vert ^{5} \vert p'_{n}(0) \vert ^{2} \vert p''_{n}(0) \vert ^{3}} \\ &\quad =4\log \rho _{n}+\log \frac{\prod_{i=1}^{3} \vert h_{n}(\xi _{0})-a_{i} \vert ^{2} \vert h'_{n}(\xi _{0})-\rho _{n}\varepsilon _{n}a_{i} \vert ^{3}}{ \vert \rho _{n}\varepsilon _{n}h_{n}(\xi _{0})-h'_{n}(\xi _{0}) \vert ^{5} \vert h'_{n}(\xi _{0}) \vert ^{2} \vert h''_{n}(\xi _{0}) \vert ^{3}}. \end{aligned}$$

(2.6)

Since $\lim_{n\to \infty}\rho _{n}\varepsilon _{n}= 0$ or ∞, we deduce

$$\begin{aligned} &\lim_{n\to \infty}\log \frac{\prod_{i=1}^{3} \vert h_{n}(\xi _{0})-a_{i} \vert ^{2} \vert h'_{n}(\xi _{0})-\rho _{n} \varepsilon _{n} a_{i} \vert ^{3}}{ \vert \rho _{n}\varepsilon _{n} h_{n}(\xi _{0})-h'_{n}(\xi _{0}) \vert ^{5} \vert h'_{n}(\xi _{0}) \vert ^{2} \vert h''_{n}(\xi _{0}) \vert ^{3})} \\ &\quad \leq \log \frac{\prod_{i=1}^{3} \vert g(\xi _{0})-a_{i} \vert ^{2}}{ \vert g'(\xi _{0}) \vert ^{-2} \vert g''(\xi _{0}) \vert ^{3}} \quad \text{as } n\to \infty . \end{aligned}$$

(2.7)

Applying Lemma 2.2 to $p_{n}(z)$ with (2.6) and (2.7), we have

$$ T(r,p_{n})\leq LD(r,p_{n};2,3,0)+O(1) \biggl(\log ^{+} \vert z_{n} \vert +m\biggl(r, \frac{p_{n}''}{p_{n}'-\varepsilon _{n} a}\biggr)+1\biggr) $$

for $0< r\leq 3$ and sufficiently large n, where $a\neq a_{j}$ ($j=1,2,3$) and $a\in C$.

By Lemmas 2.11 and 2.12 we have

$$ T(r,p_{n})\leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr). $$

In view of Lemma 2.8, we obtain

$$ T_{0}(r,p_{n})\leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr). $$

Thus we get

$$ T_{0}\bigl(3\varepsilon _{n},z_{n}+ \varepsilon _{n}\bigl(z'_{n}+ \rho _{n}\xi _{0}\bigr),f\bigr) \leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr). $$

It follows that

$$ A\bigl(2\varepsilon _{n},z_{n}+\varepsilon _{n}\bigl(z'_{n}+ \rho _{n} \xi _{0}\bigr),f\bigr) \leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr). $$

Noting that $z'_{n}+\rho _{n}\xi _{0} \to 0$, we get

$$ \bigl\{ z: \vert z-z_{n} \vert < \varepsilon _{n} \bigr\} \subseteq \bigl\{ z: \bigl\vert z-z_{n}-\varepsilon _{n}\bigl(z'_{n}- \rho _{n} \xi _{0}\bigr) \bigr\vert < 2\varepsilon _{n}\bigr\} . $$

Therefore we have

$$ A(\varepsilon _{n},z_{n},f)\leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr). $$

Case 2. Assume that $\lim_{n\to \infty}\rho _{n}\varepsilon _{n}=a_{0}\neq 0, \infty$.

Now we distinguish two subcases, $a_{0}g(z)\not \equiv g'(z)$ and $a_{0}g(z)\equiv g'(z)$.

Case 2.1. $a_{0}g(z)\not \equiv g'(z)$.

We can choose $\xi _{0}\in C$ such that

$$\begin{aligned}& g(\xi _{0})\ne 0,a_{1},a_{2},a_{3}, \infty , \qquad g'(\xi _{0})\ne 0,\infty , \\& g''( \xi _{0})\ne 0,\infty ,\qquad a_{0}g(\xi _{0})-g'( \xi _{0})\ne 0,\infty . \end{aligned}$$

Let

$$ p_{n}(z)=f_{n}\bigl(z'_{n}+ \rho _{n}\xi _{0}+z\bigr). $$

By the same arguments as in case 1, we can get

$$ A(\varepsilon _{n},z_{n},f)\leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr). $$

Case 2.2. $a_{0}g(z)\equiv g'(z)$.

We can derive that $g(z)=e^{a_{0}z+b_{0}}$, where $b_{0} \in C$. From (2.5) we obtain

$$ h_{n}(z)=f_{n}\bigl(z'_{n}+ \rho _{n}z\bigr)=f\bigl(z_{n} +\varepsilon _{n} \bigl(z'_{n}+ \rho _{n}z\bigr)\bigr)= f\bigl(z_{n} +\varepsilon _{n} z'_{n}+\varepsilon _{n} \rho _{n}z\bigr) \to g(z) $$

(2.8)

in spherical metric uniformly on compact subsets of C as $n\to \infty$,

On the other hand, noting that f and $f'$ share $a_{i}$ ($i=1,2,3$) in $A(\theta _{0},\varepsilon )$ and $f\not \equiv f'$, by Lemma 2.8 we have $S_{\theta -\varepsilon ,\theta +\varepsilon}(r,f)=O(1)$. Therefore, applying Lemma 2.9 to f in $A(\theta _{0},\varepsilon )$ we obtain

$$ \log \bigl\vert f\bigl(re^{i\phi}\bigr) \bigr\vert =r^{\omega}c\sin \bigl(\omega (\phi -\alpha )\bigr)+o \bigl(r^{ \omega}\bigr) $$

uniformly for $\theta _{0}-\varepsilon =\alpha \leq \phi \leq \beta =\theta _{0}+ \varepsilon$ as $r\notin F$ and $r\to \infty$, where c is a positive constant, $\omega =\frac{\pi}{\beta -\alpha}=\frac{\pi}{2\varepsilon}$, and F is a set of finite logarithmic measure.

Since F is a set of finite logarithmic measure, there exist a real number R ($0< R<\infty$) and a sequence of complex numbers $u_{n}$, $0<|u_{n}|<R$ for every sufficiently large n, such that

$$ \log \bigl\vert f\bigl(z_{n} +\varepsilon _{n} z'_{n}+\varepsilon _{n} \rho _{n}u_{n}\bigr) \bigr\vert =r^{ \omega}_{n} c \sin \bigl(\omega (\phi -\alpha )\bigr)+o\bigl(r^{\omega}_{n} \bigr), $$

(2.9)

where $r_{n}=|z_{n} +\varepsilon _{n} z'_{n}+\varepsilon _{n} \rho _{n}u_{n}| \notin F$, $\phi _{n}=\arg (z_{n} +\varepsilon _{n} z'_{n}+\varepsilon _{n} \rho _{n}u_{n})$, $\theta _{0}-\varepsilon /2\leq \phi \leq \theta _{0}+\varepsilon /2$, and $\alpha =\theta _{0}-\varepsilon$.

By (2.8), $h_{n}(z)=f_{n}(z'_{n}+\rho _{n}z)\to g(z)$ uniformly on $|z|\leq R$ as $n\to \infty$, and therefore $\lim_{n\to \infty} (f(z_{n} +\varepsilon _{n} z'_{n}+ \varepsilon _{n} \rho _{n} u_{n})- g(u_{n}))=0$. Noting that $u_{n}$ is a bounded sequence, there exists convergent subsequence, still denoted by $u_{n}$. Setting $u_{n}\to u_{0}$ $(n\to \infty )$, we have that $\lim_{n\to \infty}g(u_{n})=\lim_{n\to \infty}e^{a_{0}u_{n}+b_{0}}=e^{a_{0}u_{0}+b_{0}}$, so it follows that

$$ \lim_{n\to \infty} \frac {\log \vert f(z_{n} +\varepsilon _{n} z'_{n}+\varepsilon _{n} \rho _{n}u_{n}) \vert }{r^{\omega}_{n}}= 0. $$

On the other hand, by (2.8) we obtain that

$$ \lim_{n\to \infty} \frac {\log \vert f(z_{n} +\varepsilon _{n} z'_{n}+\varepsilon _{n} \rho _{n}u_{n}) \vert }{r^{\omega}_{n}}= \lim_{n\to \infty}c \sin \omega (\phi -\alpha )\geq c\sin \frac{\pi}{4}> 0. $$

We obtain a contradiction, and so case 2.2 is impossible.

This completes the proof of Lemma 2.13. □

Lemma 2.14

Let $f(z)$ be a meromorphic function with finite order $\lambda >0$, $\arg z=\theta _{0}$ be a direction, and let $\Gamma _{n}=\{z |z-z_{n}|<\varepsilon _{n}\}$ ($n=1,2,\ldots$) be a series of circles, where $z_{n}=|z_{n}|e^{i\theta _{n}}$, $\theta _{n}\to \theta _{0}$, $\lim_{n\to \infty}|z_{n}|=+\infty$, and $\varepsilon _{n}=\epsilon _{n}|z_{n}|$, $\lim_{n\to \infty}\epsilon _{n}=0$. Suppose that f and $f^{(k)}$ CM share three distinct finite complex numbers $a_{j}$ ($j=1,2,3$) in $A(\theta _{0},\varepsilon )=\{z: | \arg z-\theta _{0}|<\varepsilon \}$. If $f\not \equiv f^{(k)}$, then for every sufficiently large n ($n \geq n_{0}$),

$$ A(\varepsilon _{n},z_{n},f)\leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr), $$

(2.10)

where $\varepsilon _{n}=|z_{n}|\epsilon _{n}$.

Proof

Suppose that f and $f^{(k)}$ CM share three distinct finite complex numbers $a_{j}$ ($j=1,2,3$) in $A(\theta _{0},\varepsilon )$. Then, as in the proof of Lemma 2.13, by replacing $f'$ in Lemma 2.13 with $f^{(k)}$ and using Lemmas 2.4, 2.10, and 2.9 in $A(\theta _{0},\varepsilon )$, we can deduce (2.10). □

3 Proof of Theorem 1.1

Suppose that $f(z)\not \equiv f'(z)$. By Lemma 2.6, there exist a direction $\arg z =\theta _{0}$ and sequences $z_{n}$ and $r_{n}$ such that

$$ A\bigl(\varepsilon \bigl( \vert z_{n} \vert \bigr) \vert z_{n} \vert ,z_{n},f\bigr)\geq \frac{1}{64\pi ^{2}} \bigl\{ \beta _{p}(r_{n})\bigr\} ^{1-\frac{2}{q}} (\log r_{n})^{p-2} \quad (n=1,2, \ldots ). $$

Set $\varepsilon _{n} =|z_{n}|\varepsilon (r_{n})$, where $\varepsilon (r_{n})$ is defined in (1.2).

For arbitrary small $\varepsilon >0$, if there are three distinct complex numbers $a_{1}$, $a_{2}$, $a_{3}$ such that

$$ \overline{E}_{A(\theta _{0},\varepsilon )} (a_{j}, f)=\overline{E}_{A( \theta _{0},\varepsilon )} \bigl(a_{j}, f'\bigr),\quad j=1,2,3, $$

where $A(\theta _{0},\varepsilon )= \{z|\arg z-\theta _{0}|<\varepsilon \}$, then by Lemma 2.13 the following inequality holds:

$$ A(\varepsilon _{n},z_{n},f)\leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr), $$

(3.1)

where $|z|\leq 1$ and $\varepsilon _{n}=|z_{n}|\varepsilon (|z_{n}|)$. Combining this with (2.3), we have

$$ \beta _{p}(r_{n})^{1-\frac {2}{q}}(\log r_{n})^{p-2} \leq O(1) \bigl(1+ \log ^{+} \vert z_{n} \vert \bigr), $$

where $p\geq 3$ and $q\geq 2$.

Taking $p=3$ and noting that $|z_{n}|\le r_{n}$ and $\lim_{n\to \infty}\beta _{p}(r_{n})=\infty$, we arrive at a contradiction. This completes the proof of Theorem 1.1.

4 Proof of Theorem 1.2

Suppose that $f(z)\not \equiv f^{(k)}(z)$. By Lemma 2.6 there exist a direction $\arg z =\theta _{0}$ and sequences $z_{n}$ and $r_{n}$ such that

$$ A\bigl(\varepsilon \bigl( \vert z_{n} \vert \bigr) \vert z_{n} \vert ,z_{n},f\bigr)\geq \frac{1}{64\pi ^{2}} \bigl\{ \beta _{p}(r_{n})\bigr\} ^{1-\frac{2}{q}} (\log r_{n})^{p-2} \quad (n=1,2, \ldots ) . $$

Set $\varepsilon _{n} =|z_{n}|\varepsilon (r_{n})$, where $\varepsilon (r_{n})$ is defined in (1.2).

Next, since f and $f^{(k)}$ CM share three distinct finite complex numbers $a_{j}$ ($j=1,2,3$) in $A(\theta _{0},\varepsilon )= \{z|\arg z-\theta _{0}|<\varepsilon \}$, by Lemma 2.14 the following inequality holds:

$$ A(\varepsilon _{n},z_{n},f)\leq O(1) \bigl(1+\log ^{+} \vert z_{n} \vert \bigr), $$

(4.1)

where $|z|\leq 1$ and $\varepsilon _{n}=|z_{n}|\varepsilon (|z_{n}|)$. Combining this with (2.3), we have

$$ \beta _{p}(r_{n})^{1-\frac {2}{q}}(\log r_{n})^{p-2} \leq O(1) \bigl(1+ \log ^{+} \vert z_{n} \vert \bigr), $$

where $p\geq 3$ and $q\geq 2$.

Taking $p=3$ and noting that $|z_{n}|\le r_{n}$ and $\lim_{n\to \infty}\beta _{p}(r_{n})=\infty$, we arrive at a contradiction. This completes the proof of Theorem 1.2.