The partially shared values and small functions for meromorphic functions in a k-punctured complex plane

Abstract

The main aim of this article is to discuss the uniqueness of meromorphic functions partially sharing some values and small functions in a k-punctured complex plane Ω. We proved the following: Let $f_{1},f_{2}$ be two admissible meromorphic functions in Ω and $\alpha _{j}\ (j=1,2,\ldots ,l)$ be $l(\geq 5)$ distinct small functions with respect to f and g. If $\widetilde{E}(\alpha _{j},\varOmega ,f_{1})\subseteq \widetilde{E}(\alpha _{j},\varOmega , f_{2})\ (j=1,2,\ldots ,l)$ and

$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\overline{N} _{0} (r,\frac{1}{f_{1}-\alpha _{j}} )}{\sum_{j=1} ^{l}\overline{N}_{0} (r,\frac{1}{f_{2}-\alpha _{j}} )}> \frac{5}{2l-5}, $$

then $f_{1}\equiv f_{2}$. Our results are some improvements and extension of previous theorems given by Cao–Yi and Ge–Wu.

1 Introduction

This article is devoted to the study of uniqueness of functions which are meromorphic in a multiply-connected domain–k-punctured complex plane Ω. In 1920s, Nevanlinna gave the definition of characterized function $T(r,f)$ of meromorphic function and established the famous first and second main theorem, lemma on the logarithmic derivatives etc. of Nevalinna theory (see Hayman [1], Yang [2] and Yi and Yang [3]). Nowadays, Nevanlinna theory is a powerful tool in studying the properties of meromorphic functions in the fields of complex analysis. By applying this theory, the following well-known five-value theorem was given by Nevanlinna [4].

Theorem A

(see [4])

Iffandgare two nonconstant meromorphic functions that share five distinct values $a_{1},a_{2},a_{3},a_{4},a _{5}$IMin $X=\mathbb{C}$, then $f(z)\equiv g(z)$.

Nevanlinna [4] also pointed out the following question.

Question A

(see [4])

Does Theorem Astill hold if the five distinct values $a_{1},a_{2},a_{3}$, $a_{4},a_{5}$are replaced by five distinct small functions $\alpha _{j}\ (j=1,2,\ldots ,5)$?

Around Theorem A and Question A, the value distribution theory of meromorphic functions occupies one of the central places in complex analysis. Moreover, it is always an interesting topic how to extend and improve some important uniqueness theorems in the complex plane to the subset $\mathbb{X}$ (including the unit disc, the angular domain, the annulus, etc.). Many scholars have paid significant attention to this topic and obtained lots of meaningful and important results (see [3, 5,6,7,8,9]). For example, Fang [10] in 1999 proved the five-value theorem for meromorphic functions in the unit disc; Zheng [11] in 2003 obtained the five-value theorem for meromorphic functions in an angular domain; Cao, Yi, and Xu [12] in 2009 gave the five-value theorem for meromorphic functions in the annuli with the help of the Nevanlinna theory for meromorphic functions on annuli given by Khrystiyanyn and Kondratyuk [13, 14], or [15] in 2005, or [16] in 2004 (see [12]), etc. including [3, 10, 11, 17,18,19,20,21,22]; Yi and Yang, Lahiri, and Xu improved a series of uniqueness theorems about weight-shared and partially shared (see [3, 23,24,25,26]); there are a series of beautiful and important results related to Question A (see [27,28,29,30,31,32]). Especially, Yi [31] gave a positive answer to Question A and extended the five-value theorem to the case of sharing five distinct small functions.

Theorem B

([31] The five small functions theorem)

Letfandgbe two nonconstant meromorphic functions in a complex plane $\mathbb{C}$and $a_{j}\ (j=1,2,3,4,5)$be five distinct small functions with respect tofandg. Iffandgshare $a_{j}\ (j=1,2,3,4,5)$IM in $\mathbb{C}$, then $f\equiv g$.

In 2016, the authors investigated the uniqueness of meromorphic functions sharing some finite sets in a special multiply-connected region—k-punctured complex plane—and obtained an analog of Nevanlinna’s famous five-value theorem for meromorphic functions f and g in a k-punctured complex plane [33, 34]. To state the result, some basic notations and a definition about k-punctured complex plane should be introduced as follows, which can be found in [35].

For k distinct points $c_{j}\in \mathbb{C}$, $j\in \{1,2,\ldots ,k \}$, $\varOmega =\mathbb{C}\setminus \bigcup_{j=1}^{k}\{c_{j}\}$ can be called a k-punctured complex plane. Of course, the annulus is a special k-punctured plane as $k=1$. Let $k \geq 2$, $d=\frac{1}{2}\min \{|c _{s}-c_{j}|: j\neq s\}$, and $r_{0}=\frac{1}{d}+\max \{|c_{j}|: j \in \{1,2,\ldots ,k\}\}$, thus it yields that $\frac{1}{r_{0}}< d$,

$$ \overline{D}_{1/r_{0}}(c_{j})\cap \overline{D}_{1/r_{0}}(c_{s})=\emptyset\quad \text{for } j \neq s $$

and

$$ \overline{D}_{1/r_{0}}(c_{j})\subset D_{r_{0}}(0) \quad\text{for } j\in \{1,2,\ldots ,k\}, $$

where $D_{\delta }(c)=\{z: |z-c|<\delta \}$ and $\overline{D}_{\delta }(c)=\{z: |z-c|\leq \delta \}$. Define

$$ \varOmega _{r}=D_{r}(0)\setminus \bigcup _{j=1}^{k}\overline{D}_{1/r}(c _{j}) \quad\text{for any } r\geq r_{0}. $$

Thus, it follows that $\varOmega _{r}\supset \varOmega _{r_{0}}$ for $r_{0}< r\leq +\infty$. Obviously, $\varOmega _{r}$ is a multiple connected and $k+1$ connected region.

For a meromorphic function f in the k-punctured plane Ω and $r_{0}\leq r < +\infty$, let $n_{0}(r, f)$ denote the counting function of its poles in $\overline{\varOmega }_{r}$, and

$$\begin{aligned} &N_{0}(r,f)= \int _{r_{0}}^{r}\frac{n_{0}(t,f)}{t}\,dt, \\ &m_{0}(r,f)=\frac{1}{2\pi } \int _{0}^{2\pi }\log ^{+} \bigl\vert f\bigl(re^{i \theta }\bigr) \bigr\vert \,d\theta +\frac{1}{2\pi }\sum _{j=1}^{m} \int _{0}^{2 \pi }\log ^{+} \biggl\vert f\biggl(c_{j}+\frac{1}{r}e^{i\theta }\biggr) \biggr\vert \,d\theta \\ &\phantom{m_{0}(r,f)=}{}-\frac{1}{2\pi } \int _{0}^{2\pi }\log ^{+} \bigl\vert f\bigl(r_{0}e^{i\theta }\bigr) \bigr\vert \,d \theta - \frac{1}{2\pi }\sum_{j=1}^{m} \int _{0}^{2\pi }\log ^{+} \biggl\vert f\biggl(c _{j}+\frac{1}{r_{0}}e^{i\theta }\biggr) \biggr\vert \,d\theta , \end{aligned}$$

where $\log ^{+}x=\max \{\log x, 0\}$, then

$$ T_{0}(r,f)=m_{0}(r,f)+N_{0}(r,f) $$

is called the Nevanlinna characteristic of f in the k-punctured complex plane. Besides, we use $S(r,f)$ to denote any quantity satisfying $S(r,f) = o(T _{0}(r,f))$ for all r outside a possible exceptional set E of finite linear measure.

Definition 1.1

(see [33])

Let f be a nonconstant meromorphic function in a k-punctured plane Ω. The function f is called admissible in a k-punctured plane Ω provided that

$$ \limsup_{r\rightarrow +\infty }\frac{T_{0}(r,f)}{\log r}=+\infty ,\quad r_{0}\leq r< +\infty. $$

Remark 1.1

(see [33])

From Theorem 5 in [35], a meromorphic function f in a k-punctured plane is rational if f satisfies

$$ \limsup_{r\rightarrow +\infty }\frac{T_{0}(r,f)}{\log r}< +\infty ,\quad r_{0}\leq r< +\infty. $$

Theorem C

(see [33, Theorem 3.1])

Letfandgbe two admissible meromorphic functions inΩ; if $f,g$share five distinct values $a_{1},a_{2},a_{3},a_{4},a_{5}$IMinΩ, then $f(z)\equiv g(z)$.

2 Results

The purpose of this article is to extend and improve some uniqueness results (including Theorems A – C ) to a special multiply-connected region—k-punctured complex plane.

By relaxing the form of sharing values IM to the partially sharing in Theorem C, we obtain the first result of this article, which is an improvement of Theorem C.

Theorem 2.1

Let $f_{1},f_{2}$be two admissible meromorphic functions inΩ, $a_{1},a_{2},\ldots , a_{l}$be $l(\geq 5)$distinct values. If $\widetilde{E}(a_{j},\varOmega ,f_{1})\subseteq \widetilde{E}(a_{j}, \varOmega ,f_{2})$for all $1\leq j\leq l$and

$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\widetilde{N}_{0} (r,\frac{1}{f_{1}-a_{j}} )}{\sum_{j=1}^{l}\widetilde{N} _{0} (r,\frac{1}{f_{2}-a_{j}} )} >\frac{1}{l-3}, $$

where $\widetilde{E}(a,\varOmega ,h)=\{z| h(z)-a=0, z\in \varOmega \}$for a meromorphic function $h(z)$inΩ, where each zero is counted only once, then $f_{1}\equiv f_{2}$.

From Theorem 2.1, we can obtain the following corollary immediately.

Corollary 2.1

Let $f_{1},f_{2}$be two admissible meromorphic functions inΩ, $a_{1},a_{2},\ldots , a_{l}$be $l(\geq 5)$distinct values. If $\widetilde{E}(a_{j},\varOmega ,f_{1})\subseteq \widetilde{E}(a_{j}, \varOmega ,f_{2})$for all $1\leq j\leq l$and $f_{1}\not \equiv f_{2}$, then

$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\widetilde{N}_{0} (r,\frac{1}{f_{1}-a_{j}} )}{\sum_{j=1}^{l}\widetilde{N} _{0} (r,\frac{1}{f_{2}-a_{j}} )} \leq \frac{1}{l-3}. $$

Remark 2.1

When $l=5$ and $\widetilde{E}(a_{j},\varOmega ,f_{1})= \widetilde{E}(a _{j},\varOmega ,f_{2})$ for all $1\leq j\leq l$, in this case,

$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{5}\widetilde{N}_{0} (r,\frac{1}{f_{1}-a_{j}} )}{\sum_{j=1}^{5}\widetilde{N} _{0} (r,\frac{1}{f_{2}-a_{j}} )} =1>\frac{1}{2}. $$

Then it follows $f_{1}\equiv f_{2}$ by Theorem 2.1. Thus, this shows that Theorem 2.1 is an improvement of Theorem C.

Inspired by Question A, Theorem B, and Theorem 2.1, the second purpose of this paper is to investigate the uniqueness of meromorphic functions concerning small functions, and we obtain an analog of Nevanlinna’s five-value theorem for meromorphic functions in ak-punctured complex plane.

Theorem 2.2

Let $f_{1},f_{2}$be two admissible meromorphic functions inΩ, $\alpha _{1},\alpha _{2},\ldots , \alpha _{l}$be $l(\geq 5)$distinct small functions with respect to $f_{1},f_{2}$. If $\widetilde{E}(\alpha _{j},\varOmega ,f _{1})\subseteq \widetilde{E}(\alpha _{j},\varOmega ,f_{2})$for all $1\leq j\leq l$and

$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\widetilde{N}_{0} (r,\frac{1}{f_{1}-\alpha _{j}} )}{\sum_{j=1}^{l} \widetilde{N}_{0} (r,\frac{1}{f_{2}-\alpha _{j}} )} > \frac{5}{2l-5}, $$

where $\widetilde{E}(\alpha ,\varOmega ,h)=\{z| h(z)-\alpha (z)=0, z \in \varOmega \}$for a meromorphic function $h(z)$inΩ, where each zero is counted only once, then $f_{1}\equiv f_{2}$.

Remark 2.2

Let f be a nonconstant meromorphic function in a k-punctured plane Ω, we denote by $S(f)$ a set of meromorphic function $a(z)$ in a k-punctured plane Ω satisfying $T_{0}(r,a)=S(r,f)$, and such a meromorphic function $a(z)$ in a k-punctured plane Ω is called a small function with respect to f.

From Theorem 2.2, the following results can be obtained immediately.

Corollary 2.2

Let $f_{1},f_{2}$be two admissible meromorphic functions inΩ, and let $\alpha _{1},\alpha _{2},\ldots , \alpha _{l}$be $l(\geq 5)$distinct small functions with respect to $f_{1},f_{2}$. If $\widetilde{E}(\alpha _{j},\varOmega ,f_{1})\subseteq \widetilde{E}(\alpha _{j},\varOmega ,f_{2})$for all $1\leq j\leq l$and $f_{1}\not \equiv f _{2}$, then

$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\widetilde{N}_{0} (r,\frac{1}{f_{1}-\alpha _{j}} )}{\sum_{j=1}^{l} \widetilde{N}_{0} (r,\frac{1}{f_{2}-\alpha _{j}} )} \leq \frac{5}{2l-5}. $$

Corollary 2.3

Let $f_{1},f_{2}$be two admissible meromorphic functions inΩ, and let $\alpha _{1},\alpha _{2},\ldots , \alpha _{6}$be six distinct small functions with respect to $f_{1},f_{2}$. If $\widetilde{E}(\alpha _{j},\varOmega ,f_{1})= \widetilde{E}(\alpha _{j}, \varOmega ,f_{2})$for all $1\leq j\leq 6$, then $f_{1}\equiv f_{2}$.

3 The proof of Theorem 2.1

To prove Theorem 2.1, we require the following lemmas.

Lemma 3.1

(see [35, Theorem 3])

Let $f,f_{1},f_{2}$be meromorphic functions in ak-punctured planeΩ. Then:

• (i) the function $T_{0}(r,f)$is nonnegative, continuous, nondecreasing, and convex with respect to logron $[r_{0},+\infty )$, $T_{0}(r _{0}, f) = 0$;

• (ii) iffidentically equals a constant, then $T_{0}(r, f)$vanishes identically;

• (iii) iffis not identically equal to zero, then $T_{0}(r, f) = T _{0}(r, 1/f ), r_{0}\leq r < +\infty$;

• (iv) $T_{0}(r,f_{1}f_{2})\leq T_{0}(r,f_{1})+T_{0}(r,f_{2})+O(1)$and $T_{0}(r,f_{1}+f_{2})\leq T_{0}(r,f_{1})+T_{0}(r,f_{2})+O(1)$for $r_{0}\leq r < +\infty$;

• (v) $T_{0}(r,\frac{1}{f-a})=T_{0}(r,f)+O(1)$for any fixed $a \in \mathbb{C}$.

By using Lemma 6 in [35], we can get the following lemma easily.

Lemma 3.2

Letfbe a nonconstant meromorphic function in ak-punctured planeΩandpbe a positive integer, then

$$ m_{0}\biggl(r,\frac{f^{(p)}}{f}\biggr)=O\bigl(\log T_{0}(r,f)\bigr)+O\bigl(\log ^{+}r\bigr):=S(r,f),\quad r \rightarrow +\infty , $$

outside a setEof finite linear measure.

Remark 3.1

Obviously, if f is admissible in a k-punctured plane Ω, then

$$ m_{0}\biggl(r,\frac{f^{(p)}}{f}\biggr)=S(r,f)=o \bigl(T_{0}(r,f)\bigr). $$

Lemma 3.3

([34, Theorem 2.5])

Letfbe a nonconstant meromorphic function in ak-punctured planeΩ, and let $a_{1}, a_{2}, \ldots ,a _{q}\ (q\geq 3)$be distinct complex numbers in the extended complex plane $\widehat{\mathbb{C}}:=\mathbb{C}\cup \{\infty \}$. Then, for $r_{0}\leq r<+\infty$,

$$\begin{aligned} (q-2)T_{0}(r,f)\leq \sum_{\nu =1}^{q} \widetilde{N}_{0} \biggl(r,\frac{1}{f-a _{\nu }} \biggr)+S(r,f), \end{aligned}$$

where $\widetilde{n}_{0}(r,\frac{1}{f-a})$is the counting function of zeros of $f-a$in $\overline{\varOmega }_{r}$with the multiplicities reduced by 1,

$$ \widetilde{N}_{0}\biggl(r,\frac{1}{f-a_{\nu }}\biggr)= \int _{r_{0}}^{r}\frac{ \widetilde{n}_{0}(t,\frac{1}{f-a_{\nu }})}{t}\,dt, $$

$r\geq r_{0}$and $S(r,f)$is stated as in Lemma 3.2.

Proof of Theorem 2.1

Without loss of generality, assume that $a_{j}\ (j=1,2,\ldots ,l)$ are finite. In view of 3.3, it follows

$$\begin{aligned} (l-2)T_{0}(r,f_{1})\leq \sum _{j=1}^{l}\widetilde{N}_{0} \biggl(r, \frac{1}{f _{1}-a_{j}} \biggr)+S(r,f_{1}) \end{aligned}$$

and

$$\begin{aligned} (l-2)T_{0}(r,f_{2})\leq \sum _{j=1}^{l}\widetilde{N}_{0} \biggl(r, \frac{1}{f _{2}-a_{j}} \biggr)+S(r,f_{2}). \end{aligned}$$

Suppose that $f_{1}\not \equiv f_{2}$. In view of $\widetilde{E}(a _{j},\varOmega ,f_{1})\subseteq \widetilde{E}(a_{j},\varOmega ,f_{2})$ for all $1\leq j\leq l$, then it yields

$$ \sum_{j=1}^{l}\widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-a_{j}} \biggr) \leq \widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-f_{2}} \biggr)\leq T _{0}(r,f_{1})+T_{0}(r,f_{2})+O(1). $$

Thus, we have

$$\begin{aligned} &\sum_{j=1}^{l}\widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-a_{j}} \biggr) \\ &\quad \leq \biggl(\frac{1}{l-2}+o(1) \biggr)\sum _{j=1}^{l}\widetilde{N} _{0} \biggl(r,\frac{1}{f_{1}-a_{j}} \biggr)+ \biggl(\frac{1}{l-2}+o(1) \biggr) \sum_{j=1}^{l}\widetilde{N}_{0} \biggl(r,\frac{1}{f_{2}-a_{j}} \biggr) \end{aligned}$$

for all $r\notin E$, which implies

$$ \biggl(\frac{l-3}{l-2}+o(1) \biggr)\sum_{j=1}^{l} \widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-a_{j}} \biggr)\leq \biggl( \frac{1}{l-2}+o(1) \biggr) \sum_{j=1}^{l} \widetilde{N}_{0} \biggl(r,\frac{1}{f_{2}-a_{j}} \biggr) $$

for all $r\notin E$. Hence, it follows

$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\widetilde{N}_{0} (r,\frac{1}{f_{1}-\alpha _{j}} )}{\sum_{j=1}^{l} \widetilde{N}_{0} (r,\frac{1}{f_{2}-\alpha _{j}} )} \leq \frac{1}{l-3}, $$

which is a contradiction. Thus, $f_{1}\equiv f_{2}$.

Therefore, this completes the proof of Theorem 2.1. □

4 The proof of Theorem 2.2

To prove Theorem 2.2, we will require the following lemmas.

Lemma 4.1

Let $f_{1}(z)$and $f_{2}(z)$be two admissible meromorphic functions in ak-punctured planeΩ, $a_{t}(z)(\not \equiv 0,1)\in S(r):=S(f _{1})\cap S(f_{2})$, $t=1,2$, be not equal to constants simultaneously, and let

$$ F_{s}(z)= \begin{vmatrix} f_{s}f_{s}'&f_{s}'&f_{s}^{2}-f_{s} \\ a_{1}a_{1}'&a_{1}'&a_{1}^{2}-a_{1} \\ a_{2}a_{2}'&a_{2}'&a_{2}^{2}-a_{2} \end{vmatrix} \quad\textit{for } s=1,2. $$

(4.1)

Then:

• (i) $F_{s}(z)\not \equiv 0$for $s=1,2$.

• (ii)
  $$\begin{aligned} 2T_{0}(r,f_{s})< {}& \widetilde{N}_{0} \biggl(r,\frac{1}{f_{s}-1}\biggr)+ \widetilde{N}_{0}\biggl(r, \frac{1}{f_{s}}\biggr)+\widetilde{N}_{0}(r,f_{s})+ \widetilde{N}_{0}\biggl(r,\frac{1}{f_{s}-a_{1}}\biggr) \\ &{}+\widetilde{N}_{0}\biggl(r,\frac{1}{f_{s}-a_{2}} \biggr)+S(r,f_{1})+S(r,f_{2})\quad \textit{for } s=1,2. \end{aligned}$$

Proof

(i) Assume that $F_{1}(z)\equiv 0$. Thus, we can rewrite (4.1) as the following form:

$$ \biggl(\frac{a_{1}'}{a_{1}}-\frac{a_{2}'}{a_{2}} \biggr) \biggl(\frac{f _{1}'}{f_{1}-1}-\frac{a_{2}'}{a_{2}-1} \biggr)- \biggl( \frac{a_{1}'}{a _{1}-1}-\frac{a_{2}'}{a_{2}-1} \biggr) \biggl(\frac{f_{1}'}{f_{1}}- \frac{a _{2}'}{a_{2}} \biggr)\equiv 0. $$

(4.2)

Next, we will divide the proof into four cases as follows.

Case 1. If $\frac{a_{1}'}{a_{1}}\equiv \frac{a_{2}'}{a_{2}}$, then it follows $a_{1}=\eta _{1} a_{2}$, where $\eta _{1}\neq 1$ is a constant. From the assumptions of this lemma, it yields $\frac{a_{1}'}{a _{1}-1}\not \equiv \frac{a_{2}'}{a_{2}-1}$, which implies $\frac{f _{1}'}{f_{1}}\equiv \frac{a_{2}'}{a_{2}}$. Thus, we have $f_{1}(z)= \eta _{2}a_{2}(z)$, where $\eta _{2}$ is a constant. Therefore, we get a contradiction.

Case 2. If $\frac{a_{1}'}{a_{1}-1}\equiv \frac{a_{2}'}{a_{2}-1}$. By using the same argument as in Case 1, we also get a contradiction.

Case 3. If $\frac{a_{1}'}{a_{1}}\not \equiv \frac{a_{2}'}{a _{2}}$, $\frac{a_{1}'}{a_{1}-1}\not \equiv \frac{a_{2}'}{a_{2}-1}$ and $\frac{a_{1}'}{a_{1}}-\frac{a_{2}'}{a_{2}}\equiv \frac{a_{1}'}{a_{1}-1}-\frac{a_{2}'}{a_{2}-1}$. Thus it follows from (4.1) that

$$ \frac{f_{1}'}{f_{1}-1}-\frac{f_{1}'}{f_{1}}\equiv \frac{a_{2}'}{a_{2}-1}- \frac{a_{2}'}{a_{2}}. $$

By a simple integral, we have $\frac{1}{f_{1}}=1-\eta _{3}(1-\frac{1}{a _{2}})$, where $\eta _{3}$ is a constant, a contradiction.

Case 4. If $\frac{a_{1}'}{a_{1}}\not \equiv \frac{a_{2}'}{a _{2}}$, $\frac{a_{1}'}{a_{1}-1}\not \equiv \frac{a_{2}'}{a_{2}-1}$ and $\frac{a_{1}'}{a_{1}}-\frac{a_{2}'}{a_{2}}\not \equiv \frac{a_{1}'}{a _{1}-1}-\frac{a_{2}'}{a_{2}-1}$. Thus, we can rewrite (4.2) as the following form:

$$ \biggl(\frac{a_{1}'}{a_{1}}-\frac{a_{2}'}{a_{2}} \biggr) \frac{f_{1}'}{f _{1}-1}- \biggl(\frac{a_{1}'}{a_{1}-1}-\frac{a_{2}'}{a_{2}-1} \biggr) \frac{f _{1}'}{f_{1}} \equiv \frac{a_{1}'}{a_{1}}\frac{a_{2}'}{a_{2}-1}- \frac{a _{2}'}{a_{2}}\frac{a_{1}'}{a_{1}-1}. $$

(4.3)

By observing (4.3), the zeros of $f_{1}-1$ in Ω can only occur at the zeros, 1-points and poles of $a_{1}(z)$ and $a_{2}(z)$, and the zeros of $\frac{a_{1}'}{a_{1}}-\frac{a_{2}'}{a_{2}}$ in Ω. Thus, we have

$$\begin{aligned} \widetilde{N}_{0}\biggl(r, \frac{1}{f_{1}-1}\biggr)\leq{}& \sum_{j=1}^{2} \biggl\{ N _{0}(r,a_{j})+N_{0}\biggl(r, \frac{1}{a_{j}}\biggr)+N_{0}\biggl(r,\frac{1}{a_{j}-1} \biggr) \biggr\} \\ & {}+N_{0} \biggl(r,\frac{1}{\frac{a_{1}'}{a_{1}}-\frac{a_{2}'}{a_{2}}} \biggr) \\ ={}&S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

(4.4)

Similarly, we have

$$ \widetilde{N}_{0}\biggl(r,\frac{1}{f_{1}} \biggr)=S(r,f_{1})+S(r,f_{2}). $$

(4.5)

Further, the poles of $f_{1}$ in Ω can only occur at the zeros, 1-points and poles of $a_{1}(z)$ and $a_{2}(z)$, and the zeros of $(\frac{a_{1}'}{a_{1}}-\frac{a_{2}'}{a_{2}} )- (\frac{a _{1}'}{a_{1}-1}-\frac{a_{2}'}{a_{2}-1} )$ in Ω. By a simple calculation, we have

$$ \widetilde{N}_{0}(r,f_{1})=S(r,f_{1})+S(r,f_{2}). $$

(4.6)

By Lemma 3.3 and from (4.4)–(4.6), it follows

$$\begin{aligned} T_{0}(r,f_{1}) &< \widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-1}\biggr)+\widetilde{N} _{0}\biggl(r, \frac{1}{f_{1}}\biggr)+\widetilde{N}_{0}(r,f_{1})+S(r,f_{1}) \\ &=S(r,f_{1})+S(r,f_{2}), \end{aligned}$$

a contradiction.

If $F_{2}(z)\equiv 0$, by using the same argument as above, we also get a contradiction. Then we prove (i).

(ii) Let

$$\begin{aligned} & \delta (z)=\frac{1}{3}\min \bigl\{ 1, \bigl\vert a_{1}(z) \bigr\vert , \bigl\vert a_{2}(z) \bigr\vert , \bigl\vert a_{1}(z)-1 \bigr\vert , \bigl\vert a_{2}(z)-1 \bigr\vert , \bigl\vert a_{1}(z)-a_{2}(z) \bigr\vert , z\in \varOmega \bigr\} , \\ & \theta _{t}(r)=\bigl\{ \theta: \bigl\vert f_{1} \bigl(re^{i\theta }\bigr)-a_{t}\bigl(r^{i\theta }\bigr) \bigr\vert \leq \delta \bigl(r^{i\theta }\bigr)\bigr\} \quad (t=1,2), \\ & \theta _{3}(r)=\bigl\{ \theta: \bigl\vert f_{1} \bigl(re^{i\theta }\bigr) \bigr\vert \leq \delta \bigl(r^{i \theta } \bigr)\bigr\} , \\ & \theta _{4}(r)=\bigl\{ \theta: \bigl\vert f_{1} \bigl(re^{i\theta }\bigr)-1 \bigr\vert \leq \delta \bigl(r^{i \theta } \bigr)\bigr\} . \end{aligned}$$

Then it follows

$$\begin{aligned} & \frac{1}{2\pi } \int _{0}^{2\pi }\log \frac{1}{\delta (re^{i\theta })}\,d\theta \\ &\quad \leq\frac{1}{2\pi } \int _{0}^{2\pi }\log \max \biggl\{ 1, \frac{1}{ \vert a _{1}(z) \vert }, \frac{1}{ \vert a_{2}(z) \vert },\frac{1}{ \vert a_{1}(z)-1 \vert }, \frac{1}{ \vert a _{2}(z)-1 \vert }, \\ &\qquad\frac{1}{ \vert a_{1}(z)-a_{2}(z) \vert }, z\in \varOmega \biggr\} \,d \theta +\log 3 \\ &\quad \leq m\biggl(r,\frac{1}{a_{1}}\biggr)+m\biggl(r,\frac{1}{a_{2}} \biggr)+m\biggl(r, \frac{1}{a_{1}-1}\biggr)+m\biggl(r,\frac{1}{a_{2}-1} \biggr)\\ &\qquad{}+m\biggl(r,\frac{1}{a_{1}-a_{2}}\biggr)+4 \log 2. \end{aligned}$$

Similarly, for any $c_{j}, j=1,2,\ldots ,k$, we have

$$\begin{aligned} & \frac{1}{2\pi } \int _{0}^{2\pi }\log \frac{1}{\delta (c_{j}+ \frac{1}{r}e^{i\theta })}\,d\theta \\ &\quad\leq\frac{1}{2\pi } \int _{0}^{2\pi }\log \max \biggl\{ 1, \frac{1}{ \vert a _{1}(c_{j}+\frac{1}{r}e^{i\theta }) \vert }, \frac{1}{ \vert a_{2}(c_{j}+ \frac{1}{r}e^{i\theta }) \vert },\frac{1}{ \vert a_{1}(c_{j}+\frac{1}{r}e^{i \theta })-1 \vert }, \\ &\qquad\frac{1}{ \vert a_{2}(c_{j}+\frac{1}{r}e^{i\theta })-1 \vert },\frac{1}{ \vert a _{1}(c_{j}+\frac{1}{r}e^{i\theta })-a_{2}(c_{j}+\frac{1}{r}e^{i\theta }) \vert }, z\in \varOmega \biggr\} \,d\theta + \log 3 \\ &\quad \leq m\biggl(\frac{1}{r},\frac{1}{a_{1}(c_{j}+z)}\biggr)+m\biggl(r, \frac{1}{a_{2}(c _{j}+z)}\biggr)+m\biggl(r,\frac{1}{a_{1}(c_{j}+z)-1}\biggr) \\ &\qquad{}+m\biggl(r,\frac{1}{a_{2}(c_{j}+z)-1}\biggr)+m\biggl(r,\frac{1}{a_{1}(c_{j}+z)-a_{2}(c _{j}+z)} \biggr)+4\log 2. \end{aligned}$$

Further, for $j=1,2,\ldots ,k$,

$$ \frac{1}{2\pi } \int _{0}^{2\pi }\log \frac{1}{\delta (r_{0}e^{i\theta })}\,d\theta =O(1),\qquad \frac{1}{2\pi } \int _{0}^{2\pi }\log \frac{1}{\delta (c_{j}+\frac{1}{r _{0}}e^{i\theta })}\,d\theta =O(1). $$

Thus, it follows

$$\begin{aligned} & \frac{1}{2\pi } \int _{0}^{2\pi }\log \frac{1}{\delta (re^{i\theta })}\,d\theta +\sum_{j=1}^{k}\frac{1}{2 \pi } \int _{0}^{2\pi }\log \frac{1}{\delta (c_{j}+\frac{1}{r}e^{i \theta })}\,d\theta \\ &\qquad{} -\frac{1}{2\pi } \int _{0}^{2\pi }\log \frac{1}{\delta (r_{0}e^{i \theta })}\,d\theta -\sum_{j=1}^{k}\frac{1}{2\pi } \int _{0}^{2\pi } \log \frac{1}{\delta (c_{j}+\frac{1}{r_{0}}e^{i\theta })}\,d \theta \\ &\quad \leq m_{0}\biggl(r,\frac{1}{a_{1}} \biggr)+m_{0}\biggl(r,\frac{1}{a_{2}}\biggr)+m_{0} \biggl(r,\frac{1}{a _{1}-1}\biggr) \\ &\qquad {}+m_{0}\biggl(r,\frac{1}{a_{2}-1}\biggr)+m\biggl(r, \frac{1}{a_{1}-a_{2}}\biggr)+O(1) \\ &\quad \leq S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

(4.7)

On the other hand, taking

$$\begin{aligned} &f_{1}f_{1}'=(f_{1}-a_{1}) \bigl(f_{1}'-a_{1}' \bigr)+a_{1}'(f_{1}-a_{1})+a_{1} \bigl(f _{1}'-a_{1}' \bigr)+a_{1}a_{2}':=F_{1}, \\ &f_{1}'=\bigl(f_{1}'-a_{1}' \bigr)+a_{1}':=F_{2}, \\ &f_{1}^{2}-f_{1}=(f_{1}-a_{1})^{2}+(2a_{1}-1) (f_{1}-a_{1})+a_{1}^{2}-a _{1}:=F_{3}, \end{aligned}$$

and substituting these into (4.22), by a simple calculation, we have

$$ F= \begin{vmatrix} F_{1}-a_{1}a_{2}' & F_{2}-a_{1}' & F_{3}-a_{1}^{2}+a_{1}\\ a_{1}a_{1}' & a_{1}' & a_{1}^{2}-a_{1} \\ a_{2}a_{2}' & a_{2}' & a_{2}^{2}-a_{2} \end{vmatrix}. $$

(4.8)

From the definition of $\theta _{1}(r)$ and $\delta (z)$, we have

$$ \bigl\vert f_{1}\bigl(re^{i\theta } \bigr)-a_{1}\bigl(re^{i\theta }\bigr) \bigr\vert \leq \delta \bigl(re^{i\theta }\bigr) \leq 1+ \bigl\vert a_{1} \bigl(re^{i\theta }\bigr) \bigr\vert \quad\text{as } \theta \in \theta _{1}(r) $$

(4.9)

and

$$\begin{aligned} &\frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{F}{f_{1}-a _{1}} \biggr\vert \,d\theta \\ &\quad \leq m \biggl(r,\frac{f_{1}'-a_{1}'}{f_{1}-a_{1}} \biggr)+O\bigl(m(r,a _{1})+m(r,a_{2})+m\bigl(r,a_{1}' \bigr)+m\bigl(r,a_{2}'\bigr)\bigr) \\ &\quad< S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

(4.10)

On the other hand, we have $|f_{1}(re^{i\theta })-a_{1}(re^{i\theta })| \geq \delta (re^{i\theta })$ as $\theta \notin \theta _{1}(r)$, that is,

$$ \frac{1}{ \vert f_{1}(re^{i\theta })-a_{1}(re^{i\theta }) \vert }\leq \frac{1}{ \delta (re^{i\theta })} \quad\text{as } \theta \notin \theta _{1}(r). $$

(4.11)

By combining (4.10) and (4.11), we have

$$\begin{aligned} & m\biggl(r,\frac{1}{f_{1}-a_{1}}\biggr) \\ &\quad\leq \frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{F}{f _{1}-a_{1}} \biggr\vert \,d\theta +\frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{1}{F} \biggr\vert \,d\theta \\ &\qquad{}+\frac{1}{2\pi } \int _{[0,2\pi ]-\theta _{1}(r)}\log \biggl\vert \frac{1}{ \delta } \biggr\vert \,d\theta \\ &\quad\leq \frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{1}{F(re ^{i\theta })} \biggr\vert \,d\theta +\frac{1}{2\pi } \int _{0}^{2\pi }\log \biggl\vert \frac{1}{\delta (re^{i\theta })} \biggr\vert \,d\theta \\ &\qquad{}+S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

(4.12)

Similarly, we have

$$\begin{aligned} & m \biggl(\frac{1}{r},\frac{1}{f_{1}(c_{j}+\frac{1}{r}e^{i\theta })-a _{1}(c_{j}+\frac{1}{r}e^{i\theta })} \biggr) \\ &\quad\leq \frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{F(c _{j}+\frac{1}{r}e^{i\theta })}{f_{1}(c_{j}+\frac{1}{r}e^{i\theta }) -a _{1}(c_{j}+\frac{1}{r}e^{i\theta })} \biggr\vert \,d\theta \\ & \qquad{}+\frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+ \frac{1}{r}e^{i\theta })} \biggr\vert \,d\theta +\frac{1}{2\pi } \int _{[0,2\pi ]-\theta _{1}(r)}\log \biggl\vert \frac{1}{\delta (c_{j}+ \frac{1}{r}e^{i\theta })} \biggr\vert \,d\theta \\ &\quad\leq \frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{1}{F(c _{j}+\frac{1}{r}e^{i\theta })} \biggr\vert \,d\theta +\frac{1}{2\pi } \int _{0}^{2\pi }\log \biggl\vert \frac{1}{\delta (c_{j}+\frac{1}{r}e^{i\theta })} \biggr\vert \,d\theta \\ &\qquad{} +S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

(4.13)

Since

$$\begin{aligned} &m \biggl(r_{0},\frac{1}{f_{1}-a_{1}} \biggr)=O(1),\qquad \frac{1}{2\pi } \int _{\theta _{1}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(r_{0}e^{i\theta })} \biggr\vert \,d \theta =O(1), \\ &m \biggl(\frac{1}{r_{0}},\frac{1}{f_{1}(c_{j}+\frac{1}{r_{0}}e^{i \theta })-a_{1}(c_{j}+\frac{1}{r_{0}}e^{i\theta })} \biggr)=O(1), \end{aligned}$$

and

$$ \frac{1}{2\pi } \int _{\theta _{1}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(c _{j}+\frac{1}{r_{0}}e^{i\theta })} \biggr\vert \,d\theta =O(1), \quad j=1,2,\ldots ,k, $$

by combining (4.7), (4.12), and (4.13), it follows

$$\begin{aligned} & m_{0} \biggl(r,\frac{1}{f_{1}-a_{1}} \biggr) \\ &\quad\leq \frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{1}{F(re ^{i\theta })} \biggr\vert \,d\theta +\sum_{j=1}^{k} \frac{1}{2\pi } \int _{\theta _{1}(r)}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+\frac{1}{r}e^{i \theta })} \biggr\vert \,d\theta \\ &\qquad{}- \frac{1}{2\pi } \int _{\theta _{1}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(r _{0}e^{i\theta })} \biggr\vert \,d\theta - \sum_{j=1}^{k} \frac{1}{2\pi } \int _{\theta _{1}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+\frac{1}{r _{0}}e^{i\theta })} \biggr\vert \,d\theta \\ &\qquad{}+S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

(4.14)

By using the same argument as above, we have

$$\begin{aligned} & m_{0} \biggl(r,\frac{1}{f_{1}-a_{2}} \biggr) \\ &\quad\leq \frac{1}{2\pi } \int _{\theta _{2}(r)}\log ^{+} \biggl\vert \frac{1}{F(re ^{i\theta })} \biggr\vert \,d\theta +\sum_{j=1}^{k} \frac{1}{2\pi } \int _{\theta _{2}(r)}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+\frac{1}{r}e^{i \theta })} \biggr\vert \,d\theta \\ &\qquad{}- \frac{1}{2\pi } \int _{\theta _{2}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(r _{0}e^{i\theta })} \biggr\vert \,d\theta - \sum_{j=1}^{k} \frac{1}{2\pi } \int _{\theta _{2}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+\frac{1}{r _{0}}e^{i\theta })} \biggr\vert \,d\theta \\ &\qquad{} +S(r,f_{1})+S(r,f_{2}); \end{aligned}$$

(4.15)

$$\begin{aligned} & m_{0} \biggl(r,\frac{1}{f_{1}} \biggr) \\ &\quad\leq \frac{1}{2\pi } \int _{\theta _{3}(r)}\log ^{+} \biggl\vert \frac{1}{F(re ^{i\theta })} \biggr\vert \,d\theta +\sum_{j=1}^{k} \frac{1}{2\pi } \int _{\theta _{3}(r)}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+\frac{1}{r}e^{i \theta })} \biggr\vert \,d\theta \\ &\qquad{}- \frac{1}{2\pi } \int _{\theta _{3}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(r _{0}e^{i\theta })} \biggr\vert \,d\theta - \sum_{j=1}^{k} \frac{1}{2\pi } \int _{\theta _{3}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+\frac{1}{r _{0}}e^{i\theta })} \biggr\vert \,d\theta \\ &\qquad{}+S(r,f_{1})+S(r,f_{2}); \end{aligned}$$

(4.16)

and

$$\begin{aligned} & m_{0} \biggl(r,\frac{1}{f_{1}-1} \biggr) \\ &\quad\leq \frac{1}{2\pi } \int _{\theta _{4}(r)}\log ^{+} \biggl\vert \frac{1}{F(re ^{i\theta })} \biggr\vert \,d\theta +\sum_{j=1}^{k} \frac{1}{2\pi } \int _{\theta _{4}(r)}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+\frac{1}{r}e^{i \theta })} \biggr\vert \,d\theta \\ &\qquad{}- \frac{1}{2\pi } \int _{\theta _{4}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(r _{0}e^{i\theta })} \biggr\vert \,d\theta -\sum_{j=1}^{k} \frac{1}{2\pi } \int _{\theta _{4}(r_{0})}\log ^{+} \biggl\vert \frac{1}{F(c_{j}+\frac{1}{r _{0}}e^{i\theta })} \biggr\vert \,d\theta \\ &\qquad{}+S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

(4.17)

Since $\theta (r)\in [0,2\pi )$, then from (4.14)–(4.17) it yields

$$\begin{aligned} &m_{0} \biggl(r,\frac{1}{f_{1}-a_{1}} \biggr)+m_{0} \biggl(r,\frac{1}{f _{1}-a_{2}} \biggr)+m_{0} \biggl(r,\frac{1}{f_{1}} \biggr)+m_{0} \biggl(r, \frac{1}{f _{1}-1} \biggr) \\ &\quad < m_{0} \biggl(r,\frac{1}{F} \biggr)+S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

(4.18)

If $z_{0}$ is a zero of $f_{1}$ or $f_{1}-1$ or $f_{1}-a_{1}$ or $f_{1}-a_{2}$ in Ω of multiplies $p>1$ and not a pole of $a_{1}$ or $a_{2}$ in Ω, then $z_{0}$ must be a zero of $F_{1}(z)$ in Ω of multiplies $p-1$. Thus, it follows

$$\begin{aligned} 4T_{0}(r,f_{1})< {}&N_{0}\biggl(r, \frac{1}{f_{1}}\biggr)+N_{0}\biggl(r,\frac{1}{f_{1}-1} \biggr)+N _{0}\biggl(r,\frac{1}{f_{1}-a_{1}}\biggr)+N_{0} \biggl(r,\frac{1}{f_{1}-a_{2}}\biggr) \\ &{}-N_{0}\biggl(r,\frac{1}{F_{1}}\biggr)+T_{0}(r,F_{1})+O(1)+S(r,f_{1})+S(r,f_{2}) \\ < {}&\widetilde{N}_{0}\biggl(r, \frac{1}{f_{1}}\biggr)+\widetilde{N}_{0}\biggl(r, \frac{1}{f _{1}-1}\biggr)+\widetilde{N}_{0}\biggl(r, \frac{1}{f_{1}-a_{1}}\biggr)+\widetilde{N}_{0}\biggl(r, \frac{1}{f _{1}-a_{2}}\biggr) \\ &{}+T_{0}(r,F_{1})+O(1)+S(r,f_{1})+ S(r,f_{2}). \end{aligned}$$

(4.19)

In addition, from the definition of $F_{1}(z)$, we can get

$$\begin{aligned} &m_{0}(r,F_{1})< 2m_{0}(r,f_{1})+S(r,f_{1})+S(r,f_{2}), \end{aligned}$$

(4.20)

$$\begin{aligned} &N_{0}(r,F_{1})< 2N_{0}(r,f_{1})+ \widetilde{N}_{0}(r,f_{1})+S(r,f_{1})+S(r,f _{2}). \end{aligned}$$

(4.21)

Hence, from (4.19)–(4.21), we can get Lemma 4.1(ii).

Therefore, this completes the proof of Lemma 4.1. □

Lemma 4.2

Let $f_{1}(z)$and $f_{2}(z)$be two admissible meromorphic functions in ak-punctured planeΩ, $\alpha _{j}(z)(\not \equiv 0,1) \in S(f_{1})\cap S(f_{2})$, $j=1,2,\ldots ,5$, be five distinct meromorphic functions in ak-punctured planeΩ, then

$$ 2T_{0}(r,f_{s})< \sum _{j=1}^{5} \widetilde{N}_{0} \biggl(r,\frac{1}{f_{s}- \alpha _{j}}\biggr)+S(r,f_{1})+S(r,f_{2}),\quad s=1,2. $$

(4.22)

Proof

Set

$$\begin{aligned} &g_{s}=\frac{f_{s}-\alpha _{4}}{f_{s}-\alpha _{5}}\frac{\alpha _{3}- \alpha _{5}}{\alpha _{3}-\alpha _{4}}\quad (s=1,2), \\ &a_{j}=\frac{\alpha _{j}-\alpha _{4}}{\alpha _{j}-\alpha _{5}}\frac{ \alpha _{3}-\alpha _{5}}{\alpha _{3}-\alpha _{4}}\quad (j=1,2). \end{aligned}$$

Then it yields

$$\begin{aligned} & \bigl\vert T_{0}(r,g_{s})-T_{0}(r,f_{s}) \bigr\vert < S(r,f_{1})+S(r,f_{2}),\quad \text{for } s=1,2, \end{aligned}$$

(4.23)

$$\begin{aligned} &S(r,f_{1})+S(r,f_{2})=S(r,g_{1})+S(r,g_{2}). \end{aligned}$$

(4.24)

Here we will consider three cases as follows.

Case 1. If $g_{1}$ and $g_{2}$ are admissible, then by applying Lemma 4.1 for $g_{1},g_{2},a_{1},a_{2}$, we have

$$\begin{aligned} 2T_{0}(r,g_{s})< {} & \widetilde{N}_{0}(r,g_{s})+\widetilde{N}_{0} \biggl(r,\frac{1}{g _{s}}\biggr)+\widetilde{N}_{0}\biggl(r, \frac{1}{g_{s}-1}\biggr)+\widetilde{N}_{0}\biggl(r, \frac{1}{g _{s}-a_{1}}\biggr) \\ &{}+\widetilde{N}_{0}\biggl(r,\frac{1}{g_{s}-a_{2}} \biggr)+S(r,g_{1})+ S(r,g_{2}) \end{aligned}$$

(4.25)

for $s=1,2$. From the notations of $g_{1}$ and $g_{2}$ and $\alpha _{j}\in S(f_{1})\cap S(f_{2})$, we have

$$\begin{aligned} &\widetilde{N}_{0}(r,g_{s})< N_{0} \biggl(r,\frac{1}{f_{s}-\alpha _{5}} \biggr)+O \Biggl(\sum _{j=1}^{5}T_{0}(r,\alpha _{j}) \Biggr), \end{aligned}$$

(4.26)

$$\begin{aligned} &\widetilde{N}_{0}\biggl(r,\frac{1}{g_{s}} \biggr)< N_{0} \biggl(r,\frac{1}{f_{s}- \alpha _{4}} \biggr)+O \Biggl(\sum _{j=1}^{5}T_{0}(r,\alpha _{j}) \Biggr), \end{aligned}$$

(4.27)

and

$$ \widetilde{N}_{0}\biggl(r,\frac{1}{g_{s}-a_{j}} \biggr)< N_{0} \biggl(r,\frac{1}{f _{s}-\alpha _{j}} \biggr)+O \Biggl(\sum _{j=1}^{5}T_{0}(r,\alpha _{j}) \Biggr) $$

(4.28)

for $s=1,2; j=1,2$. Substituting (4.26)–(4.28) into (4.25), we can get (4.22) easily.

Case 2. If $g_{1}$ and $g_{2}$ are rational, then from Remark 2.1 we have $T_{0}(r,g_{s})=O(\log r)=S(r,f_{1})+S(r,f_{2})$ for $s=1,2$. Thus, combining (4.23) and (4.24), it yields $T_{0}(r,f_{s})=S(r,f_{1})+S(r,f_{2})$ for $s=1,2$. Hence the conclusions hold.

Case 3. If one of $g_{1},g_{2}$ is rational, without loss of generality assume that $g_{1}$ is rational and $g_{2}$ is admissible. From Case 2 and Case 1, we have $T_{0}(r,f_{1})=S(r,f_{1})+S(r,f_{2})$ and

$$\begin{aligned} T_{0}(r,f_{2})< \sum_{j=1}^{5} \widetilde{N}_{0}\biggl(r,\frac{1}{f_{2}- \alpha _{j}}\biggr)+S(r,f_{1})+S(r,f_{2}). \end{aligned}$$

Then the conclusion holds.

From Cases 1–3, this completes the proof of Lemma 4.2. □

Proof of Theorem 2.2

Take any distinct $s_{1}, \ldots ,s_{5}\in \{1,2,\ldots ,l\}$, and in view of Lemma 4.2, it follows

$$\begin{aligned} &2T_{0}(r,f_{i})< \sum _{j=1}^{5} \widetilde{N}_{0} \biggl(r,\frac{1}{f_{i}- \alpha _{s_{j}}}\biggr)+S(r),\quad i=1,2. \end{aligned}$$

(4.29)

Thus, we can conclude

$$ 2 \begin{pmatrix} l \\ 5 \end{pmatrix}T_{0}(r,f_{i})\leq \frac{5}{l} \begin{pmatrix} l \\ 5 \end{pmatrix}\sum_{j=1}^{l} \widetilde{N}_{0} \biggl(r,\frac{1}{f_{i}-\alpha _{j}} \biggr)+S(r),\quad i=1,2, $$

that is,

$$ T_{0}(r,f_{i})\leq \frac{5}{2l}\sum_{j=1}^{l} \widetilde{N}_{0} \biggl(r,\frac{1}{f _{i}-\alpha _{j}} \biggr)+S(r),\quad i=1,2. $$

(4.30)

Suppose that $f_{1}\not \equiv f_{2}$. In view of $\widetilde{E}( \alpha _{j},\varOmega ,f_{1})\subseteq \widetilde{E}(\alpha _{j},\varOmega ,f _{2})$ for all $1\leq j\leq l$, we have

$$ \sum_{j=1}^{l} \widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-\alpha _{j}} \biggr) \leq \widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-f_{2}} \biggr)\leq T _{0}(r,f_{1})+T_{0}(r,f_{2})+O(1). $$

(4.31)

In view of (4.30) and (4.31), we can deduce

$$\begin{aligned} &\sum_{j=1}^{l}\widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-\alpha _{j}} \biggr) \\ &\quad \leq \biggl(\frac{5}{2l}+o(1) \biggr)\sum _{j=1}^{l}\widetilde{N} _{0} \biggl(r,\frac{1}{f_{1}-\alpha _{j}} \biggr) + \biggl(\frac{5}{2l}+o(1) \biggr) \sum_{j=1}^{l}\widetilde{N}_{0} \biggl(r,\frac{1}{f_{2}-\alpha _{j}} \biggr) \end{aligned}$$

for $r\notin E$, which implies

$$ \biggl(\frac{2l-5}{2l}+o(1) \biggr)\sum_{j=1}^{l} \widetilde{N}_{0} \biggl(r,\frac{1}{f_{1}-\alpha _{j}} \biggr)\leq \biggl( \frac{5}{2l}+o(1) \biggr) \sum_{j=1}^{l} \widetilde{N}_{0} \biggl(r,\frac{1}{f_{2}-\alpha _{j}} \biggr) $$

for $r\notin E$. This leads to

$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\widetilde{N}_{0} (r,\frac{1}{f_{1}-\alpha _{j}} )}{\sum_{j=1}^{l} \widetilde{N}_{0} (r,\frac{1}{f_{2}-\alpha _{j}} )} \leq \frac{5}{2l-5}, $$

which is a contradiction with the assumption of Theorem 2.2. Thus, $f_{1}\equiv f_{2}$.

Therefore, this completes the proof of Theorem 2.2. □