Meromorphic Functions Sharing Three Values with Their Derivatives in an Angular Domain

Abstract

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

1 INTRODUCTION AND MAIN RESULT

Let $f:C\rightarrow\hat{C}=C\bigcup\{\infty\}$ be a meromorphic function, where $C$ is the complex plane. It is assumed that the reader is familiar with the basic results and notations of the 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 $\mu$ and the order $\lambda$ of a meromorphic function $f$ are in turn defined as follows

$$\mu:=\mu(f)=\liminf_{r\to\infty}\frac{\log T(r,f)}{\log r},$$

$$\lambda:=\lambda(f)=\limsup_{r\to\infty}\frac{\log T(r,f)}{\log r}.$$

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

In 1979, Gundersen [5] and Mues and Steinmetz [10] have considered the uniqueness of a meromorphic function $f$ and its derivative $f^{\prime}$ 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 $f^{\prime}$ share $a_{j}(j=1,2,3)$ $IM$, then $f\equiv f^{\prime}$.

Later on, Frank and Schwick [3] generalized the above results 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 $f^{(k)}$ share $a,b,c$ $IM$, 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 $a\in C\bigcup\{\infty\}$ and an integer $p\geq 0$. Let the pairs of real numbers $\{\alpha_{j},\beta_{j}\}(j=1,...,q)$ be such that

$$-\pi\leq\alpha_{1}<\beta_{1}\leq\alpha_{2}<\beta_{2}\leq...\leq\alpha_{q}<\beta_{q}\leq\pi,$$

with $\omega=\max\left\{\dfrac{\pi}{\beta_{j}-\alpha_{j}}:1\leq j\leq q\right\}$ , and

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

where $\delta=\max\{\omega,\mu\}$ . For a positive integer $k$ , assume that $f$ and $f^{(k)}$ share three distinct finite complex numbers $a_{j}(j=1,2,3)$ $IM$ 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 et al. [9] observed that Theorem C is invalid for $q\geq 2$ and proved the following more general result, which extends Theorem C (see [9, p. 443]).

Theorem D:  (see [9]).  Let $f$ be a transcendental meromorphic function of finite lower order $\mu(f)$ in C and 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

$$-\pi\leq\alpha_{1}<\beta_{1}\leq\alpha_{2}<\beta_{2}\leq...\leq\alpha_{q}<\beta_{q}\leq\pi$$

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

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

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

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

(1.1)

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

Theorem E:   Let $f$ be a nonconstant meromorphic function of lower order $\mu(f)>1/2$ in $C$, $a_{j}(j=1,2,3)$ 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 $L[f]$ share $a_{j}(j=1,2,3)$ $CM$ in D, then $f=L[f]$.

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

$$\bar{E}_{D}(f,a_{j})=\bar{E}_{D}(f^{\prime},a_{j}),\quad j=1,2,3$$

for some typical set in $C$ in steads of 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 $f(z)=e^{2z}$, it is clear that $f(z)\neq f^{\prime}(z)$, but $|f(z)|$ is bounded by $1$ on $D$ being the left half plane. Thus

$$\bar{E}_{D}(f,n)=\bar{E}_{D}(f^{\prime},n)=\emptyset\quad\text{for any}\quad n>1.$$

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

Based on the theory on singular direction for a meromorphic function (see [14]) and the research results of shared values of a meromorphic function (see [8, 12]), combining with the result of Theorems D and E, we may conjecture that angular domain of the singular direction may be the right. The main result of this paper shows that it is true when $D$ is an angular domain with the Borel direction as the center line for $f$ with order $\lambda>0$, which extends Theorems D and E.

In order to prove our main results, we introduce some notations about Ahlfors–Shimizu character of meromorphic function in $C$.

$$T_{0}(r,f)=\int\limits_{0}^{r}\frac{A(t)}{t}dt,\quad A(t)=\frac{1}{\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{t}\left(\frac{|f^{\prime}(\rho e^{i\theta})|}{1+|f(\rho e^{i\theta})|^{2}}\right)^{2}d\rho d\theta.$$

(1.2)

We recall the Nevanlinna theory on an angular domain.

Let $f$ be a meromorphic function in $D=\{z:\alpha\leq\arg z\leq\beta\}$, where $0\leq\beta-\alpha\leq 2\pi$. Nevanlinna [11] defined the following symbols (also see [4]).

$$A_{\alpha,\beta}(r,f)=\frac{\omega}{\pi}\int\limits_{1}^{r}\left(\frac{1}{t^{\omega}}-\frac{t^{\omega}}{r^{2\omega}}\right)\{\log^{+}|f(te^{i\alpha})|+\log^{+}|f(te^{i\beta})|\}\frac{dt}{t},$$

$$B_{\alpha,\beta}(r,f)=\frac{2\omega}{\pi r^{\omega}}\int\limits_{\alpha}^{\beta}\log^{+}|f(re^{i\theta})|\sin\omega(\theta-\alpha)d\theta,$$

$$C_{\alpha,\beta}(r,f)=2\sum_{1<|b_{m}|<r}\left(\frac{1}{|b_{m}|^{\omega}}-\frac{|b_{m}|^{\omega}}{r^{2\omega}}\right)\sin\omega(\theta_{m}-\alpha),$$

$$S_{\alpha,\beta}(r,f)=A_{\alpha,\beta}(r,f)+B_{\alpha,\beta}(r,f)+C_{\alpha,\beta}(r,f)$$

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

Throughout the paper, we denote by $R(r,*)$ a quantity satisfying

$$R(r,*)=O\{\log(rT(r,*))\},r\in E,$$

where $E$ denotes a set of positive real numbers with finite linear measure, which will not necessarily be the same in each occurrence. To state our result, we need the following theorem F and definitions .

Theorem F  (see [7]). Let $f$ be a meromorphic function of infinite order in $C$. Then there exists a function $\rho(r)$ such that

• (i) $\rho(r)$ is continuous and nondecreasing for $r\geq r_{0}$ , and $\rho(r)\to+\infty$ as $r\to+\infty$ .

• (ii) $U(r)=r^{\rho(r)}(r\geq r_{0})$ satisfies the condition

  $$\lim\limits_{r\to+\infty}\frac{\log U(R)}{\log U(r)}=1,\quad R=r+\frac{r}{\log U(r)}.$$

• (iii) $\limsup\limits_{r\to\infty}\frac{\log T(r,f)}{\rho(r)\log r}=1$.

The function $\rho(r)$ is also called the precise order of $f$.

Definition 1.1 (see [13]). Let $f$ be a meromorphic function of finite order $\lambda(f)>0$ in $C$. A direction $\arg z=\theta_{0}$ ($0\leq\theta_{0}<2\pi$) is called a Borel direction of $f(z)$ of order $\lambda(f)$ if for arbitrary small positive $\varepsilon$ the following relation holds:

$$\lim\limits_{r\to\infty}\frac{\log n(r,\theta_{0},\varepsilon,f=a)}{\log r}=\lambda(f)$$

for all $a\in\hat{C}=C\bigcup+\infty$ except at most two exceptional values, where $n(r,\theta_{0},\varepsilon,f=a)$ denotes the number of the zeros of $f-a$ counting multiplicities in the sector $|\arg z-\theta_{0}|<\varepsilon$,$|z|\leq r$.

Definition 1.2 (see [7]). Let $f$ be a meromorphic function of infinite order in $C$ and let $\rho(r)$ be the precise order of $f$. A direction $\arg z=\theta_{0}$ ($0\leq\theta_{0}<2\pi$) is called a Borel direction of $f(z)$ with precise order $\rho(r)$ if for arbitrary small positive $\varepsilon$ the following relation holds:

$$\lim\limits_{r\to\infty}\frac{\log n(r,\theta_{0},\varepsilon,f=a)}{\rho(r)\log r}=1$$

for all $a\in{\hat{C}}$ except at most two exceptional values, where $n(r,\theta_{0},\varepsilon,f=a)$ is as in Definition 1.1.

In this paper we will prove the following theorem.

Theorem 1.1. Let $f$ be a meromorphic function of finite order $\lambda(f)>0$ in $C$ and $\varepsilon$ be an arbitrary small positive number, and a direction $\arg z=\theta_{0}$ ($0\leq\theta_{0}<2\pi$) be a Borel direction of $f(z)$. Assume that $f$ and $f^{\prime}$ share three distinct finite complex numbers $a_{j}(j=1,2,3)$ $IM$ in $A(\theta_{0},\varepsilon)$, where $A(\theta_{0},\varepsilon)=\{z:|\arg z-\theta_{0}|<\varepsilon\}$. Then $f\equiv f^{\prime}$.

Theorem 1.2. Let $f$ be a meromorphic function of infinite order in $C$ and a direction $\arg z=\theta_{0}$ ($0\leq\theta_{0}<2\pi$) be a Borel direction of $f(z)$ with precise order $\rho(r)$. Then for arbitrary positive number $\varepsilon$, $f$ and $f^{\prime}$ share two finite values IM at most in the angular region $\{z:|\arg z-\theta_{0}|<\varepsilon\}$.

Theorem 1.3. Let $f$ be a meromorphic function of infinite order in ${C}$ and $L[f]$ defined by (1.1), and $\arg z=\theta_{0}$ ($0\leq\theta_{0}<2\pi$) be a Borel direction of $f(z)$ with precise order $\rho(r)$. Then for arbitrary positive $\varepsilon$, $f$ and $L[f]$ share two finite values $CM$ at most in the angular region $\{z:|\arg z-\theta_{0}|<\varepsilon\}$.

2 PRELIMINARY

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

Lemma 2.1 ([1, 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 ${f_{n}}\in\mathcal{F}$;

• (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 and, moreover, $g$ is of order at most two.

For convenience, we will use the following notation

$$LD(r,f:c_{1},c_{2})=c_{1}\left[m\left(r,\frac{f^{\prime}}{f}\right)+\sum_{i=1}^{3}m\left(r,\frac{f^{\prime}}{f-a_{i}}\right)\right]+c_{2}\left[m\left(r,\frac{f^{\prime\prime}}{f^{\prime}}\right)+\sum_{i=1}^{3}m\left(r,\frac{f^{\prime\prime}}{f^{\prime}-ta_{i}}\right)\right].$$

Lemma 2.2 ([12]). Let $f$ be a meromorphic function in a domain $D=\{z:|z|<R\}$ and $a_{j}(j=1,2,3)$ be three distinct finite complex numbers, and let $t$ be a positive real number and $a\in C$. If

$$\bar{E}_{D}(a_{j},f)=\bar{E}_{D}(ta_{j},f^{\prime})\quad for\quad j=1,2,3;$$

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

$$T(r,f)\leq LD(r,f:2,3)+\log\frac{\prod_{i=1}^{3}|f(0)-a_{i}|^{2}|f^{\prime}(0)-ta_{i}|^{3}}{|tf(0)-f^{\prime}(0)|^{5}|f^{\prime}(0)|^{2}}$$

$${}+3\log\frac{1}{|f^{\prime\prime}(0)|}+\left(\log^{+}t+m\left(r,\frac{f^{\prime\prime}}{f^{\prime}-ta}\right)+1\right)O(1).$$

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 ([14]). Let $f(z)$ be a meromorphic function with finite order $\lambda>0$ and $\arg z=\theta_{0}$ is a Borel direction of $f$. Then there exist a series of circles

$$\Gamma_{j}=\{z:|z-z_{j}|<\epsilon_{j}|z_{j}|\},$$

where $z_{j}=|z_{j}|e^{i\theta_{0}},$ $\lim\limits_{j\to\infty}|z_{j}|=+\infty,\lim\limits_{j\to\infty}\epsilon_{j}=0(j=1,2,\cdot\cdot\cdot)$, such that $f$ takes any complex number at least $|z_{j}|^{\lambda-\delta_{j}}$ times in every circle $\Gamma_{j}$ with at most some exceptional values contained in two circles with spherical radius $2^{-j}$, where $\lim\limits_{j\to\infty}|\delta_{j}|=0$.

Lemma 2.4 ([14]). Let $\mathcal{F}$ be a family of meromorphic function on domain $D$, then $\mathcal{F}$ is normal on $D$ if and only if, for every bounded closed domain $K\subseteq D$, there exists a positive number $M$ such that every $f\in\mathcal{F}$

$$\frac{|f^{\prime}(z)|}{1+|f(z)|^{2}}\leq M.$$

Lemma 2.5 ([6, 17]). Let $m$ be the normalized area measure on the Riemann sphere $S$. Then we have

$$A(r,f)=\int\limits_{\hat{C}}n(r,f=a)dm(a),$$

where ${\hat{C}}=C\bigcup\{\infty\}.$

Lemma 2.6 ([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

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

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

Lemma 2.7 ([8]). Let $f(z)$ be a nonconstant meromorphic function in the complex plane, and $a_{1}$, $a_{2}$, $a_{3}$ are three distinct finite complex numbers. Assume that $f$ and $f^{\prime}$ share the $a_{i}(i=1,2,3)$ $IM$ 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) $f\equiv f^{\prime}$, or (ii) $S_{\alpha,\beta}(r,f)=Q(r,f)$, where $Q(r,f)$ is such a quantity 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 $Q(r,f)=O(\log(rT(r,f))$ for $r\notin E$ and $r\to\infty$ and $E$ denotes a set of positive real numbers with finite linear measure.

Lemma 2.8 ([4, 9]). Let $f$ be a meromorphic function on $\overline{\Omega}(\alpha,\beta)$. If $S_{\alpha,\beta}(r,f)=O(1)$, then

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

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

Lemma 2.9 ([13]). Let $f$ be a meromorphic function of infinite order in $C$, and let $\rho(r)$ be a precise order of $f$. Then a direction $\arg z=\theta_{0}$ is a Borel direction of precise order $\rho(r)$ of $f$ if and only if for arbitrarily small $\varepsilon>0$ we have

$$\limsup_{r\to+\infty}\frac{\log S_{\theta_{0}-\varepsilon,\theta_{0}+\varepsilon}(r,f)}{\rho(r)\log r}=1.$$

Lemma 2.10 ([2]). Let $f$ be a meromorphic function of infinite order in $C$, $a_{j}(j=1,2,3)$ be three distinct finite complex numbers and let $L[f]$ be given by (1.1). Suppose that $f$ and $L[f]$ share $a_{j}(j=1,2,3)$ $CM$ in $D=\{z:\alpha\leq\arg z\leq\beta\}$, where $0<\beta-\alpha\leq 2\pi$. If $f\not\equiv L[f]$, then $S_{\alpha,\beta}(r,f)=R(r,f)$.

Lemma 2.11 ([14]). Let $f(z)$ be a meromorphic function in disk $D(0,R)$ centered at $0$ with radius $R$. If $f(0)\neq 0,\infty,$ then we have for $0<r<\rho<R$

$$m(r,\frac{f^{(k)}}{f})<c_{k}\left\{1+\log^{+}\log^{+}\left|\frac{1}{f(0)}\right|+\log^{+}\frac{1}{r}+\log^{+}\frac{1}{\rho-r}+\log^{+}\rho+\log^{+}T(\rho,f)\right\},$$

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 $a(r)$ 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 $b,c$.

Lemma 2.13. Let $f(z)$ be a meromorphic function with finite order $\lambda>0$ and $\arg z=\theta_{0}$ be a Borel direction of $f$, and $\Gamma_{j}=\{z:|z-z_{j}|<\epsilon_{j}|z_{j}|\}$ be a series of circles, where $z_{j}=|z_{j}|e^{i\theta_{0}},$ and$\lim\limits_{j\to\infty}|z_{j}|=+\infty,\lim\limits_{j\to\infty}\epsilon_{j}=0(j=1,2,\cdot\cdot\cdot)$. Suppose that $f$ and $f^{\prime}$ share three distinct finite complex numbers $a_{j}(j=1,2,3)$ $IM$ in $A(\theta_{0},\varepsilon)$, where $A(\theta_{0},\varepsilon)=\{z:|\arg z-\theta_{0}|<\varepsilon\}$. If $f\not\equiv f^{\prime}$, then for every sufficiently large $n(n\geq n_{0})$,

$$A(\varepsilon_{n},z_{n},f)\leq O(1)(1+\log^{+}|z_{n}|),$$

(2.1)

where $\varepsilon_{n}=|z_{n}|\epsilon_{n}$.

Proof. Set $f_{n}(z)=f(z_{n}+\varepsilon_{n}z)$. We consider two cases:

Case 1. Assume that $f_{n}(z)$ be normal at $|z|\leq 1$, by Lemma 2.4, implying that

$$\frac{|f^{\prime}_{n}(z)|}{1+|f_{n}(z)|^{2}}=\frac{\varepsilon_{n}|f^{\prime}(z_{n}+\varepsilon_{n}z)|}{1+|f(z_{n}+\varepsilon_{n}z)|^{2}}\leq M\quad(n=1,2,...)$$

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

$$A(\varepsilon_{n},z_{n},f)=\frac{1}{\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\varepsilon_{n}}\left(\frac{|f^{\prime}(z_{n}+\rho e^{i\theta})|}{1+|f(z_{n}+\rho e^{i\theta})|^{2}}\right)^{2}\rho d\rho d\theta\leq 2M^{2}.$$

So, (2.1) holds.

Case 2. Assume that $f_{n}(z)$ is not normal at $|z|\leq 1$.

According to Lemma 2.1, there exist

(1) a sequence of point $\{z^{\prime}_{n}\}\subset\{|z|<1\}$;

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

(3) positive numbers $\rho_{n}$ with $\rho_{n}\to 0(n\to\infty)$; such that

$$h_{n}(z)=f_{n}(z^{\prime}_{n}+\rho_{n}z)\to g(z)$$

(2.2)

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

$$h_{n}^{(k)}(\xi)={\rho_{n}}^{k}f_{n}^{(k)}(z^{\prime}_{n}+\rho_{n}\xi)\to g^{(k)}(\xi).$$

We claim $g^{\prime\prime}(\xi)\not\equiv 0$. Otherwise, $g(z)=cz+d,\ (c,d\in\textbf{C}$ 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

$$h_{n}(\xi_{n})=f_{n}(z^{\prime}_{n}+\rho_{n}\xi_{n})=g(\xi_{0})=a_{1}.$$

Notice that $f$ and $f^{\prime}$ share $a_{1}$ IM in $\{z:|\arg z-\theta_{0}|<\varepsilon\}$, we have

$$c=g^{\prime}(\xi_{0})=\lim_{n\to\infty}h^{\prime}_{n}(\xi_{n})=\lim_{n\to\infty}\rho_{n}\varepsilon_{n}f^{\prime}(z_{n}+\varepsilon_{n}(z^{\prime}_{n}+\rho_{n}\xi_{n}))$$

$${}=\lim_{n\to\infty}\rho_{n}\varepsilon_{n}f(z_{n}+\varepsilon_{n}(z^{\prime}_{n}+\rho_{n}\xi_{n}))=\lim_{n\to\infty}\rho_{n}\varepsilon_{n}a_{1}.$$

Thus, we have

$$\lim_{n\to\infty}\rho_{n}\varepsilon_{n}=\frac{c}{a_{1}}.$$

For finite complex number $a_{2}$, we can choose $\eta_{0}$ with $g(\eta_{0})=a_{2}$. By Hurwitz’s theorem, there exists a sequence $\eta_{n}\to\eta_{0}$ such that

$$h_{n}(\eta_{n})=f_{n}(z^{\prime}_{n}+\rho_{n}\eta_{n})=g(\eta_{0})=a_{2}.$$

Likewise, we get

$$\lim_{n\to\infty}\rho_{n}\varepsilon_{n}=\frac{c}{a_{2}},$$

this gives a contradiction.

For a sequence of positive numbers $\rho_{n}\varepsilon_{n}$, it is easy to know that there exists a subsequence, we still denoted by $\rho_{n}\varepsilon_{n}$, such that $\lim\limits_{n\to\infty}\rho_{n}\varepsilon_{n}=a_{0}$, where $a_{0}\in[0,+\infty)\bigcup\{+\infty\}$. Now, we consider two cases: $a_{0}=0$ or $+\infty$ and $0<a_{0}<+\infty$.

Case 2.1. Assume that $\lim\limits_{n\to\infty}\rho_{n}\varepsilon_{n}=0$ or $\infty$.

We choose $\xi_{0}\in C$ such that

$$g(\xi_{0})\neq 0,a_{1},a_{2},a_{3},\infty,g^{\prime}(\xi_{0})\neq 0,\infty,g^{\prime\prime}(\xi_{0})\neq 0,\infty.$$

Let $p_{n}(z)=f_{n}(z^{\prime}_{n}+\rho_{n}\xi_{0}+z)$ for arbitrary small $\varepsilon>0$, in view of

$$\overline{E}_{A(\theta_{0},\varepsilon)}(a_{j},f)=\overline{E}_{A(\theta_{0},\varepsilon)}(a_{j},f^{\prime}),\quad j=1,2,3,$$

and $\lim\limits_{n\to\infty}\epsilon_{n}=0$, and for sufficiently large $n$,

$$\Gamma_{n}=\{z|z-z_{n}|<\epsilon_{n}|z_{n}|,z_{n}=|z_{n}|e^{i\theta_{0}}\}\subseteq A(\theta_{0},\varepsilon/2).$$

Therefore, for every sufficiently large $n(n\geq n_{0})$, we have

$$\bar{E}_{D}(a_{i},p_{n}(z))=\bar{E}_{D}(\varepsilon_{n}a_{i},p^{\prime}_{n}(z))\quad(i=1,2,3),$$

where $D=\{z:|z|<4\}$. Note that

$$p_{n}(0)=f_{n}(z^{\prime}_{n}+\rho_{n}\xi_{0})=h_{n}(\xi_{0})\to g(\xi_{0})\neq a_{1},a_{2},a_{3},\infty,$$

$$p^{\prime}_{n}(0)=f^{\prime}_{n}(z^{\prime}_{n}+\rho_{n}\xi_{0})=\frac{h^{\prime}_{n}(\xi_{0})}{\rho_{n}},\quad h^{\prime}_{n}(\xi_{0})\to g^{\prime}(\xi_{0}),$$

$$p^{\prime\prime}_{n}(0)=f^{\prime\prime}_{n}(z^{\prime}_{n}+\rho_{n}\xi_{0})=\frac{h^{\prime\prime}_{n}(\xi_{0})}{\rho_{n}^{2}},\quad h^{\prime\prime}_{n}(\xi_{0})\to g^{\prime\prime}(\xi_{0}),$$

$$\varepsilon_{n}p_{n}(0)-p^{\prime}_{n}(0)=\frac{\varepsilon_{n}\rho_{n}h_{n}(\xi_{0})-h^{\prime}_{n}(\xi_{0})}{\rho_{n}}.$$

Thus, we have

$$\log\frac{\prod_{i=1}^{3}|p_{n}(0)-a_{i}|^{2}|p^{\prime}_{n}(0)-\varepsilon_{n}a_{i}|^{3}}{|\varepsilon_{n}p_{n}(0)-p^{\prime}_{n}(0)|^{5}|p^{\prime}_{n}(0)|^{2}}+3\log\frac{1}{|p^{\prime\prime}_{n}(0)|}$$

$${}=\log\frac{\displaystyle\prod_{i=1}^{3}|p_{n}(0)-a_{i}|^{2}|p^{\prime}_{n}(0)-\varepsilon_{n}a_{i}|^{3}}{|\varepsilon_{n}p_{n}(0)-p^{\prime}_{n}(0)|^{5}|p^{\prime}_{n}(0)|^{2}|p^{\prime\prime}_{n}(0)|^{3}}$$

$${}=4\log\rho_{n}+\log\frac{\displaystyle\prod_{i=1}^{3}|h_{n}(\xi_{0})-a_{i}|^{2}|h^{\prime}_{n}(\xi_{0})-\rho_{n}\varepsilon_{n}a_{i}|^{3}}{|\rho_{n}\varepsilon_{n}h_{n}(\xi_{0})-h^{\prime}_{n}(\xi_{0})|^{5}|h^{\prime}_{n}(\xi_{0})|^{2}|h^{\prime\prime}_{n}(\xi_{0})|^{3}}.$$

(2.3)

Since $\lim\limits_{n\to\infty}\rho_{n}\varepsilon_{n}=0$ or $\infty$, by simple calculation we can deduce for sufficiently large $n(n\geq n_{0})$

$$\log\frac{\prod_{i=1}^{3}|h_{n}(\xi_{0})-a_{i}|^{2}|h^{(k)}_{n}(\xi_{0})-\rho^{k}_{n}\varepsilon_{n}a_{i}|^{3}}{|\rho^{k}_{n}\varepsilon_{n}h_{n}(\xi_{0})-h^{(k)}_{n}(\xi_{0})|^{5}|h^{(k)}_{n}(\xi_{0})|^{2}|h^{(k+1)}_{n}(\xi_{0})|^{3}}\leq{O(1)\log^{+}|z_{n}|}.$$

(2.4)

Applying Lemma 2.2 to $p_{n}(z)$ with properties (2.3), (2.4), we have

$$T(r,p_{n})\leq LD(r,p_{n};2,3)+O(1)\left(\log^{+}|z_{n}|+m\left(r,\frac{p_{n}^{\prime\prime}}{p_{n}^{\prime}-\varepsilon_{n}a}\right)+1\right)$$

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

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

In view of Lemma 2.6, we obtain

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

Thus, we get

$$T_{0}(3\varepsilon_{n},z_{n}+\varepsilon_{n}(z^{\prime}_{n}+\rho_{n}\xi_{0}),f)\leq O(1)(1+\log^{+}|z_{n}|).$$

It follows that

$$A(2\varepsilon_{n},z_{n}+\varepsilon_{n}(z^{\prime}_{n}+\rho_{n}\xi_{0}),f)\leq O(1)(1+\log^{+}|z_{n}|).$$

Note that $z^{\prime}_{n}+\rho_{n}\xi_{0}\to 0$, we get

$$\{z:|z-z_{n}|<\varepsilon_{n}\}\subseteq\{z:|z-z_{n}-\varepsilon_{n}(z^{\prime}_{n}-\rho_{n}\xi_{0})|<2\varepsilon_{n}\}.$$

Therefore, we have

$$A(\varepsilon_{n},z_{n},f)\leq O(1)(1+\log^{+}|z_{n}|).$$

Case 2.2. Assume that $\lim\limits_{n\to\infty}\rho_{n}\varepsilon_{n}=a_{0},a_{0}\neq 0,\infty.$ Now, we distinguish two subcases $a_{0}g(z)\not\equiv g^{\prime}(z)$ and $a_{0}g(z)\equiv g^{\prime}(z)$.

Case 2.2.1. $a_{0}g(z)\not\equiv g^{\prime}(z)$. We can choose $\xi_{0}\in C$ such that

$$g(\xi_{0})\neq 0,a_{1},a_{2},a_{3},\infty,g^{\prime}(\xi_{0})\neq 0,\infty,g^{\prime\prime}(\xi_{0})\neq 0,\infty,a_{0}g(\xi_{0})-g^{\prime}(\xi_{0})\neq 0,\infty.$$

Let

$$p_{n}(z)=f_{n}(z^{\prime}_{n}+\rho_{n}\xi_{0}+z).$$

By the same arguments as in case 2.1, we can get

$$A(\varepsilon_{n},z_{n},f)\leq O(1)(1+\log^{+}|z_{n}|).$$

Case 2.2.2. $a_{0}g(z)\equiv g^{\prime}(z)$. We can derive that $g(z)=e^{a_{0}z+b_{0}}$, where $b_{0}\in C$. From (2.2), we obtain

$$h_{n}(z)=f_{n}(z^{\prime}_{n}+\rho_{n}z)=f(z_{n}+\varepsilon_{n}(z^{\prime}_{n}+\rho_{n}z))=f(z_{n}+\varepsilon_{n}z^{\prime}_{n}+\varepsilon_{n}\rho_{n}z)\to g(z).$$

(2.5)

On the other hand, noting that $f$ and $f^{\prime}$ share $a_{i},i=1,2,3$ in $A(\theta_{0},\varepsilon)$, by Lemma 2.7, we have $S_{\theta-\varepsilon,\theta+\varepsilon}(r,f)=O(1)$. Therefore, applying Lemma 2.8 to $f$ in $A(\theta_{0},\varepsilon)$, we obtain

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

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, $\omega=\dfrac{\pi}{\beta-\alpha}=\dfrac{\pi}{2\varepsilon}$, and $F$ is a set of finite logarithmic measure.

Noting that $F$ is a set of finite logarithmic measure. Therefore, 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

$$\log|f(z_{n}+\varepsilon_{n}z^{\prime}_{n}+\varepsilon_{n}\rho_{n}u_{n})|=r^{\omega}_{n}c\sin(\omega(\phi-\alpha))+o(r^{\omega}_{n}),$$

(2.6)

where $r_{n}=|z_{n}+\varepsilon_{n}z^{\prime}_{n}+\varepsilon_{n}\rho_{n}u_{n}|\notin F$, $\phi_{n}=\arg(z_{n}+\varepsilon_{n}z^{\prime}_{n}+\varepsilon_{n}\rho_{n}u_{n})$, $\theta_{0}-\varepsilon/2\leq\phi_{n}\leq\theta_{0}+\varepsilon/2$, and $\alpha=\theta_{0}-\varepsilon$.

From (2.5), we get $\lim\limits_{n\to\infty}(f(z_{n}+\varepsilon_{n}z^{\prime}_{n}+\varepsilon_{n}\rho_{n}u_{n})-g(u_{n}))=0$. Noting that $u_{n}$ is a bounded sequence, there exists convergent subsequence, we still denote it by $u_{n}$ and set $u_{n}\to u_{0}(n\to\infty)$. We have $\lim\limits_{n\to\infty}g(u_{n})=\lim\limits_{n\to\infty}e^{a_{0}u_{n}+b_{0}}=e^{a_{0}u_{0}+b_{0}}$, and it follows that

$$\lim\limits_{n\to\infty}\frac{\log|f(z_{n}+\varepsilon_{n}z^{\prime}_{n}+\varepsilon_{n}\rho_{n}u_{n})|}{r^{\omega}_{n}}=0.$$

On the other hand, by (2.6) we obtain that

$$\lim\limits_{n\to\infty}\frac{\log|f(z_{n}+\varepsilon_{n}z^{\prime}_{n}+\varepsilon_{n}\rho_{n}u_{n})|}{r^{\omega}_{n}}=\lim\limits_{n\to\infty}c\sin\omega(\phi-\alpha)\geq c\sin\frac{\pi}{4}>0$$

we obtain a contradiction, and so case 2.2 is false. This completes the proof of Lemma 2.13.

3 PROOF OF THEOREMS

Proof of Theorem 1.1. Suppose that $f\not\equiv f^{\prime}$, since $\arg z=\theta_{0}$ is a Borel direction of $f$, by Lemma 2.3, there exists a series of circles

$$\Gamma_{j}=\{z:|z-z_{j}|<\epsilon_{j}|z_{j}|\},$$

where $z_{j}=|z_{j}|e^{i\theta_{0}},$ and $\lim\limits_{j\to\infty}|z_{j}|=+\infty,\lim\limits_{j\to\infty}\epsilon_{j}=0(j=1,2,\cdot\cdot\cdot),$ such that $f$ takes any complex number at least $|z_{j}|^{\lambda-\delta_{j}}$ times in every circle $\Gamma_{j}$ with at most some exceptional values contained in two circles with spherical radius $2^{-j}$, where $\lim\limits_{j\to\infty}|\delta_{j}|=0$. We denote the two circles by $\Delta_{j1}$ and $\Delta_{j2}$.

Therefore, by Lemma 2.5, we have

$$A(\epsilon_{j}|z_{j}|,z_{j},f)=\int\limits_{\hat{C}}n(\epsilon_{j}|z_{j}|,z_{j},f=a)dm(a)$$

$${}\geq\int\limits_{\hat{C}-\Delta_{j1}-\Delta_{j2}}n(\epsilon_{j}|z_{j}|,z_{j},f=a)dm(a)\geq\frac{1}{2}|z_{j}|^{\lambda-\delta_{j}}.$$

(III.1)

On the other hand, from Lemma 2.13, the following inequality holds

$$A(\varepsilon_{n},z_{n},f)\leq O(1)(1+\log^{+}|z_{n}|),$$

(3.2)

where $|z|\leq 1$ and $\varepsilon_{n}=|z_{n}|\epsilon_{n}$.

Combining with (3.1) and (3.2), we get

$$\frac{1}{2}|z_{n}|^{\lambda-\delta_{n}}\leq A(\varepsilon_{n},z_{n},f)\leq O(1)(1+\log^{+}|z_{n}|).$$

Noting that $\lambda>0$ and $\lim\limits_{n\to\infty}\delta_{n}=0$, this contradicts with $\lim\limits_{n\to\infty}|z_{n}|=+\infty$. The proof of Theorem 1.1 is complete.

Proof of Theorem 1.2. Suppose that $f$ and $f^{\prime}$ share three distinct finite complex numbers $a_{j}(j=1,2,3)$ $IM$ in $A(\theta_{0},\varepsilon)$, by Lemma 2.7, in view of $f$ with infinite order and $f\not\equiv f^{\prime}$, we have $S_{\theta_{0}-\varepsilon,\theta_{0}+\varepsilon}(r,f)=R(r,f)$, implying that

$$S_{\theta_{0}-\varepsilon,\theta_{0}+\varepsilon}(r,f)=O(\log U(r)),U(r)=r^{\rho(r)}.$$

On the other hand, $\arg z=\theta_{0}$ is a Borel direction of $f$ with precise order $\rho(r)$. By Lemma 2.9, for arbitrarily small $\varepsilon>0$, we have

$$\limsup_{r\to+\infty}\frac{\log S_{\theta_{0}-\varepsilon,\theta_{0}+\varepsilon}(r,f)}{\rho(r)\log r}=1.$$

Thus, we arrive at a contradiction. This completes the proof of Theorem 1.2.

Proof of Theorem 1.3. Suppose that $f$ and $L[f]$ share three distinct finite complex numbers $a_{j}(j=1,2,3)$ $CM$ in $A(\theta_{0},\varepsilon)$. Using Lemmas 2.10 and 2.9 in $A(\theta_{0},\varepsilon)$, similar to proof of Theorem 1.2, we can conclude a contradiction. This completes the proof of Theorem 1.3.