Uniqueness of meromorphic functions sharing small functions in the k-punctured complex plane

1.   Introduction

As is known to all, Nevanlinna value distribution is a powerful tool in studying the properties of meromorphic functions in the fields of complex analysis. In 1926, R. Nevanlinna gave the definition of characterise 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 ^[5], Yang ^[20] and Yi and Yang ^[21]). Moreover, the two main theorems occupy a central place in the value distribution theory of meromorphic functions. By using these results, R. Nevanlinna in 1926 proved the following well-known five-values theorem and gave a problem about small function.

Theorem 1.1. (see ^[21]). If $f$ and $g$ are two non-constant meromorphic functions that share five distinct values $a_1, a_2, a_3, a_4, a_5$ $IM$ in $X = \mathbb{C}$, then $f(z)\equiv g(z)$.

Question 1.1. (see ^[21]). Does Theorem 1.1 still hold if the five distinct values $a_1, a_2, a_3$, $a_4, a_5$ are replaced by five distinct small functions $a_j\; (j = 1, 2, \ldots, 5)$?

After this theorem and question, many mathematicians had paid considerate attention to the uniqueness of meromorphic functions with shared values in the whole complex plane (see ^[21]). There were a series of beautiful uniqueness theorems about value-shared and small functions shared (see ^{[10,11,17,18,22,23]}). Especially, Yi ^[22] gave a positive answer to Question 1.1 and extend the five value theorem to the case of sharing five distinct small functions.

Theorem 1.2. (see ^[22] (The five small functions theorem)) Let $f$ and $g$ be two non-constant meromorphic functions in complex plane $\mathbb{C}$, and $a_{j} (j = 1, 2, 3, 4, 5)$ be five distinct small functions with respect to $f$ and $g$. If $f$ and $g$ share $a_{j} (j = 1, 2, 3, 4, 5)$ IM in $\mathbb{C}$, then $f\equiv g$.

In the past several decades, it is an increasing interest to investigate the uniqueness of meromorphic functions on a subset of complex plane $\mathbb{C}$, such as: the unit disc, the angular domain and the whole complex plane, and a lot of important theorems were obtained (see ^[21]). In 1999, Fang ^[3] discussed the uniqueness of admissible functions sharing some sets in the unit disc. Around 2003, Zheng in ^[24,25] studied the uniqueness problem and obtained five-values theorem and four-values theorem in some angular domain of $\mathbb{C}$. In 2015, Liu and Mao ^[12] further extended Theorem 1.2 to an angular domain and obtained the following theorem.

Theorem 1.3. (see [12,Theorem 1.1]). Let $f$ and $g$ be two nonconstant meromorphic functions in an angular domain $\mathbb{X}(\alpha, \beta): = \{z: \alpha < \arg z < \beta\}$ with $0 < \beta-\alpha\leq 2\pi$ such that

and let $a_j (j = 1, \ldots, 5)$ be five distinct small functions with respect to $f$ and $g$ in $\mathbb{X}(\alpha, \beta)$. If $f$ and $g$ share $a_j (j = 1, \ldots, 5)$ $IM$ in $\mathbb{X}(\alpha, \beta)$, then $f\equiv g$.

As we all know, the whole complex plane, unit disc and angular domain can all be regarded as a simply connected region. Of late, with the establishment of Nevanlinna theory for meromorphic functions on annuli given by Khrystiyanyn and Kondratyuk ^[6,7] or ^[8] in 2005 or ^[9] in 2004, it is very interesting to consider the uniqueness of meromorphic function on doubly connected regions (see ^[1,2,13,14]). In 2009 and 2011, Cao ^[1,2] investigated the uniqueness of meromorphic functions on annuli sharing some values and some sets, and obtained an analog of Nevanlinna's famous five-values theorem.

Theorem 1.4. (see [1,Thereom 3.2] or [2,Corollary 3.3]). Let $f$ and $g$ be two transcendental or admissible meromorphic functions on the annulus $\mathbb{A} = \{z: \frac{1}{R_{0}} < |z| < R_{0}\},$ where $1 < R_{0}\leq+\infty.$ Let $a_{j}$ $(j = 1, 2, 3, 4, 5)$ be five distinct complex numbers in $\overline{\mathbb{C}}.$ If $f$ and $g$ share $a_j$ $IM$ for $j = 1, 2, 3, 4, 5$, then $f(z)\equiv g(z).$

In 2015, Wu and Ge ^[14] extended Theorem 1.4 when five values are replaced by five small functions and obtained

Theorem 1.5. (see [14,Theorem 1.1]). Let $f$ and $g$ be two transcendental or admissible meromorphic functions on the annulus $\mathbb{A} = \{z: \frac{1}{R_0} < |z| < R_0\}$, where $1 < R_0\leq+\infty$. Let $a_j(j = 1, 2, \ldots, 5)$ be five distinct small functions with respect to $f$ and $g$ on the annulus $\mathbb{A}$. If $f$ and $g$ share $a_j(j = 1, 2, \ldots, 5)$ IM, then $f(z)\equiv g(z)$.

Very recently, the authors ^[16] investigated the uniqueness of meromorphic function in a special multiply connested region-$k$-punctured complex plane, and obtained an analog of Nevanlinna's famous five-values theorem for meromorphic functions $f$ and $g$ in a $k$-punctured complex plane, and gave a question as follows.

Theorem 1.6. (see [16,Theorem 3.2]). Let $f$ and $g$ be two admissible meromorphic functions in $\Omega$, if $f, g$ share five distinct values $a_1, a_2, a_3, a_4, a_5$ $IM$ in $\Omega$, then $f(z)\equiv g(z)$.

Question 1.2. (see [16,Remark 3.3]). Does the conclusion of Theorem 1.6 still holds if $a_j(j = 1, \ldots, 5)$ are replaced by small functions $a_j(z)(j = 1, \ldots, 5)$.

2.   Results

Motivated by Question 1.2 and Theorems 1.2–1.6, the main purpose of this article is to investigate the uniqueness of meromorphic functions concerning small functions, and we obtain an analog of Nevanlinna's famous five-values theorem for meromorphic functions in $k$-punctured complex plane. To state our main results, let us recall some basic notations about function-shared in a $k$-punctured plane as follows.

Let $f$ be a non-constant meromorphic function in a $k$-punctured plane $\Omega$, we denote $S(f)$ a set of meromorphic function $a(z)$ in a $k$-punctured plane $\Omega$ satisfying $T_0(r, a) = S(r, f)$, and such a meromorphic function $a(z)$ in a $k$-punctured plane $\Omega$ is called as a small function with respect to $f$.

Let $f$ be a non-constant meromorphic function in a $k$-punctured plane $\Omega$, a small function $\alpha(z)\in S(f)\cup\{\infty\}$ and a positive integer $l$ (or $+\infty$). We use $E_\Omega(\alpha, l;f)$ to denote the set of zeros of $f(z)-\alpha$ in $\Omega$ of multiplicity less than $l$ (counting the multiplicities), and $\widetilde{E}_\Omega(\alpha, l;f)$ to denote the set of zeros of $f(z)-\alpha(z)$ in $\Omega$ of multiplicity less than $l$ (ignoring the multiplicities), especially, $E_\Omega(\alpha, l;f) = E_\Omega(\alpha, f)$ and $\widetilde{E}_\Omega(\alpha, l;f) = \widetilde{E}_\Omega(\alpha, f)$ as $l = \infty$.

For two non-constant meromorphic functions $f$ and $g$ in $\Omega$ and $\alpha\in (S(f)\cap S(g))\cup\{\infty\}$, if $E_\Omega(\alpha, f) = E_\Omega(\alpha, g)$, then we say that $f$ and $g$ share $\alpha$ $CM$ in $\Omega$, if $\widetilde{E}_\Omega(\alpha, f) = \widetilde{E}_\Omega(\alpha, g)$, then we say that $f$ and $g$ share $\alpha$ $IM$ in $\Omega$.

The following inequality concerning five small functions plays a key role in proving our main theorem.

Theorem 2.1. Let $f$ and $g$ be two transcendental or admissible meromorphic functions in $\Omega$, $\alpha_j(z)\in S(f)\cap S(g)\; (j = 1, 2, \ldots, 5)$ be five distinct small functions. If $l$ is a positive integer $\geq 4$, $f$ and $g$ satisfy

Then we have $S(r): = S(r, f) = S(r, g)$,

and

where $\widetilde{n}_0(r, \alpha_j; f|\leq l)$ is the counting function of distinct poles of $\frac{1}{f-\alpha_j}$ in $\overline{\Omega}_r$ with the multiplicities less than $l$,

By applying the above result, we obtain the main theorem below, which gives a positive answer to Question 1.2.

Theorem 2.2. Let $f$ and $g$ be two transcendental or admissible meromorphic functions in $\Omega$, $\alpha_j\in S(f)\cap S(g), j = 1, 2, \ldots, 5$ be five distinct small functions and $l$ a positive integer. If $\widetilde{E}_\Omega(\alpha_j, l; f) = \widetilde{E}_\Omega(\alpha_j, l; g)$ for $j = 1, 2, \ldots, 5$, and $l\geq 22$, then $f(z)\equiv g(z)$.

Remark 2.1. By comparing with Theorem 2.2 and Theorem 1.6, we can see that five constants have been replaced by five small functions, and $IM$ shared has been also relaxed to partial weighted shared (only under the condition that the distinct zeros of $f-\alpha_j$ and $g-\alpha_j$ in $\Omega$ of multiplicity less than $l$ are same for $j = 1, 2, \ldots, 5$). Hence, Theorem 2.2 is an extension and improvement of Theorem 1.6.

Remark 2.2. In view of Theorem 2.2, two naturel questions arise:

Question 2.1. What condition on $l_j, (j = 1, 2, , \ldots, 5)$ can guarantee that the conclusion of Theorem 2.2 still holds if $\widetilde{E}_\Omega(\alpha_j, l; f) = \widetilde{E}_\Omega(\alpha_j, l; g)$ for $j = 1, 2, \ldots, 5$ are replaced by $\widetilde{E}_\Omega(\alpha_j, l_j; f) = \widetilde{E}_\Omega(\alpha_j, l; g)$ for $j = 1, 2, \ldots, 5$, and $l_j$ are five positive integers?

Question 2.2. Does the conclusion of Theorem 2.2 still hold, if the $k$-punctured complex plane $\Omega$ is turned to a more general domain

where $G_0$ is a bounded simply connected domain, and $\{K_j\}, K_j\subset G_0$, $j = 1, 2, \ldots, m$, are connected compacts not degenerating to a point and such that $G_j = \mathbb{C}\backslash K_j$ is a domain, $j = 1, 2, \ldots, m$?

Moreover, similar to the argument as in the proof of Theorem 2.2, we can easily get the following result, which is an improvement of Theorem 1.4.

Corollary 2.1. Let $f$ and $g$ be two transcendental or admissible meromorphic functions on the annulus $\mathbb{A} = \{z: \frac{1}{R_0} < |z| < R_0\}$, where $1 < R_0\leq+\infty$. Let $a_j(j = 1, 2, \ldots, 5)$ be five distinct small functions with respect to $f$ and $g$ on the annulus $\mathbb{A}$. If $\widetilde{E}_\mathbb{A}(a_j, l; f) = \widetilde{E}_\mathbb{A}(a_j, l; g)$ for $j = 1, 2, \ldots, 5$, and $l\geq 22$, then $f(z)\equiv g(z)$.

When $l = +\infty$, one can get the following conclusion.

Theorem 2.3. Let $f$ and $g$ be two transcendental or admissible meromorphic functions in $\Omega$, $\alpha_j\in S(f)\cap S(g), (j = 1, 2, \ldots, 5)$ be five distinct small functions. If $\widetilde{E}_\Omega(\alpha_j, f) = \widetilde{E}_\Omega(\alpha_j, g)$, for $(j = 1, 2, \ldots, 5)$, then $f(z)\equiv g(z)$.

Remark 2.3. Theorem 2.3 can be called as 5 IM theorem for meromorphic functions in $k$-punctured complex plane.

3.   Nevanlinna theory in k-punctured complex planes

Let $k$ be a positive integer, for $k$ distinct points $c_j\in \mathbb{C}$, $j\in\{1, 2, \ldots, k\}$, we say that $\Omega = \mathbb{C}\setminus\bigcup_{j = 1}^k\{c_j\}$ is a $k$-punctured complex plane. The main purpose of this article is to discuss meromorphic functions in those $k$-punctured planes for which $k \geq 2$.

Let $d = \frac{1}{2}\min\{|c_s-c_j|: j\neq s\}$ and $r_0 = \frac{1}{d}+\max\{|c_j|: j\in \{1, 2, \ldots, k\}\}$, thus it follows $\frac{1}{r_0} < d$ and

and

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

Thus, it yields that $\Omega_r\supset\Omega_{r_0}$ for $r_0 < r\leq +\infty$. Moreover, it is easy to see that $\Omega_r$ is $k+1$ connected region.

In 2007, Hanyak and Kondratyuk ^[4] gave some extension of the Nevanlinna value distribution theory for meromorphic functions in a $k$-punctured complex plane and proved a series of theorems which is an analog of the result on the whole plane $\mathbb{C}$.

Let $f$ be a meromorphic function in a $k$-punctured plane $\Omega$, we denote $n_0(r, f)$ to be the counting function of its poles in $\overline{\Omega}_r$, $r_0\leq r < +\infty$ and

and we also define

where $\log^{+}x = \max\{\log x, 0\}$ and $r_0\leq r < +\infty$, then

is called as the Nevanlinna characteristic of $f$. 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 of finite linear measure.

Theorem 3.1. (see [4,Theorem 3]). Let $f, f_1, f_2$ be meromorphic functions in a $k$-punctured plane $\Omega$. Then

(i) the function $T_0(r, f)$ is non-negative, continuous, non-decreasing and convex with respect to $\log r$ on $[r_0, +\infty)$, $T_0(r_0, f) = 0$;

(ii) if $f$ identically equals a constant, then $T_0(r, f)$ vanishes identically;

(iii) if $f$ is not identically equal to zero, then $T_0(r, f) = T_0(r, 1/f), r_0\leq r < +\infty$;

(iv) $T_0(r, f_1f_2)\leq T_0(r, f_1)+T_0(r, f_2)+O(1)$ 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 $a \in \mathbb{C}$.

Remark 3.1. Theorem 3.1 (i)–(iv) show the elementary properties of meromorphic function $f(z)$ in the $k$-punctured plane $\Omega$, Theorem 3.1 (v) can be said the Jensen-Nevanlinna formula for meromorphic function $f(z)$ in the $k$-punctured plane $\Omega$, which is another expression of Jensen's formula and exhibits the relations between characteristic functions of $f$ and $\frac{1}{f}$ in $\Omega$. Theorem 3.1 (v) is also called as the first fundamental theorem of the value distribution theory in the $k$-punctured plane.

Definition 3.1. Let $f$ be a nonconstant meromorphic function in $k$-punctured plane $\Omega$. The function $f$ is called admissible in $k$-punctured plane $\Omega$ provided that

Remark 3.2. From Theorem 5 in ^[4], we have that a meromorphic function $f$ in $k$-punctured plane is rational if $f$ satisfies

By using Lemma 6 in ^[4], we can get the following lemma easily.

Theorem 3.2. Let $f$ be a nonconstant meromorphic function in a $k$-punctured plane $\Omega$, and $p$ a positive integer, then

outside a set of finite linear measure.

Remark 3.3. Obviously, if $f$ is admissible in a $k$-punctured plane $\Omega$, then

outside a set of finite linear measure.

Remark 3.4. This is an important result which is often used in this paper.

4.   The proof of Theorem 2.1

To prove Theorem 2.1, we require the following lemma.

Lemma 4.1. (see [15,Lemma 5.3.2]). Let $f_1(z)$ and $f_2(z)$ be two admissible meromorphic functions in a $k$-punctured plane $\Omega$, $\alpha_j(z)(\not\equiv0, 1)\in S(f_1)\cap S(f_2)$ $j = 1, 2, \ldots, 5$ be five distinct meromorphic functions in a $k$-punctured plane $\Omega$, then

The proof of Theorem 2.1: In view of the assumptions of Theorem 2.1, it follows

that is,

Similarly, we have

From (4.2) and (4.3), we can deduce immediately that $S(r) = S(r, f) = S(r, g)$.

By applying Lemma 4.1 for $f(z)$ and $\alpha_j(z) (j = 1, 2, \ldots, 5)$ again, we have

where $\widetilde{n}_0(r, \alpha_j; f|\geq l+1)$ is the counting function of distinct poles of $\frac{1}{f-\alpha_j}$ in $\overline{\Omega}_r$ with the multiplicities great than $l+1$,

Thus, it follows

Similarly, we have

By Lemma 4.1 and (4.4), it yields

that is,

Similarly, we have

Thus, from the above two inequalities, it follows

And since $\widetilde{E}_\Omega(\alpha_j, l; f) = \widetilde{E}_\Omega(\alpha_j, l; g), (j = 1, 2, \ldots, 5)$, that is, $\widetilde{N}_0(r, \alpha_j; f|\leq l)+\widetilde{N}_0(r, \alpha_j; g|\leq l)$ for $j = 1, 2, \ldots, 5$, then we can prove (2.3) easily by combining (4.6).

Thus, we complete the proof of Theorem 2.1.

5.   The proof of Theorem 2.2

To prove Theorem 2.2, some lemmas below will be required.

Lemma 5.1. (see ^[16]). Let $f$ be a nonconstant meromorphic function in $k$-punctured plane $\Omega$, and let

be an irreducible rational function in $f$ with coefficients $\{a_i\}$ and $\{b_j\}$, where $a_n\neq0$ and $b_m\neq0$. Then

where $d = max\{n, m\}$.

Lemma 5.2. Let $f$ be a transcendental or admissible meromorphic functions in $\Omega$, $a(z), b(z)\in S(f)$ be two distinct small functions with respect to $f$. Set

Then we have $L(f, a, b)\not\equiv0$ and for $i = 0, 1$,

Proof. Utilizing the determinant nature, we obtain that

which further yields

Suppose that $L(f, a, b)\equiv0$, that is,

by a simply integral, we get $f = \frac{1}{1-\eta}(b-\eta a)$, where $\eta$ is a constant, which is a contradiction with $a(z), b(z)\in S(f)$. So, we have $L(f, a, b)\not\equiv0$. If $i = 0$, by applying Theorem 3.2 for (5.1), it follows

If $i = 1$, we have

By combining $a(z), b(z)\in S(f)$ and applying Theorem 3.2 for (5.2), we get

Therefore, this completes the proof of Lemma 5.2.

Let $h_1(z)$ and $h_2(z)$ be two non-constant meromorphic functions in $\Omega$ and $\alpha(z)$ (or $\infty$) be the common small function of $h_1(z)$ and $h_2(z)$, we use $\widetilde{N}_0(r, h_1(z) = \alpha(z) = h_2(z))$ ($\widetilde{N}^E_0(r, h_1(z) = \alpha(z) = h_2(z))$) to denote the counting function of those common zeros of $h_1(z)-\alpha(z)$ and $h_2(z)-\alpha(z)$ in $\Omega$, regardless of multiplicity (with the same multiplicity), and each zeros counted only once. Moreover, if $\widetilde{N}_0(r, \frac{1}{h_j(z)-\alpha(z)})-\widetilde{N}^E_0(r, h_1(z) = \alpha(z) = h_2(z)) = S(r, h_j)$, $j = 1, 2$, then we can say that $h_1(z)$ and $h_2(z)$ share $\alpha(z)$ $CM^*$; if $\widetilde{N}_0(r, \frac{1}{h_j(z)-\alpha(z)})-\widetilde{N}_0(r, h_1(z) = \alpha(z) = h_2(z)) = S(r, h_j)$, $j = 1, 2$, then we can say that $h_1(z)$ and $h_2(z)$ share $\alpha(z)$ $IM^*$.

Lemma 5.3. Let $f$ and $g$ be two transcendental or admissible meromorphic functions in $\Omega$, $b_j(z)\in S(f)\cap S(g)\; (j = 1, 2, 3)$. If $f, g$ share $b_j(z) (j = 1, 2, 3)$ and $\infty$ $IM^*$. Set

Then $T_0(r, \mathcal{H}_1) = S(r, f)+S(r, g)$.

Proof. By applying Lemma 5.2 for $\mathcal{H}_1$, we can obtain

Now, we will estimate $N_0(r, \mathcal {H}_1)$. We know that the poles of $\mathcal {H}_1$ in $\Omega$ can come from the zeros of $f-b_j(z), g-b_j(z)$ in $\Omega$ for $j = 1, 2, 3$, the poles of $f, g$ in $\Omega$. Because $\mathcal {H}_1$ can be represented as

and since $f, g$ share $b_j(z)$ $IM^*$ in $\Omega$ and $b_j(z)\in S(f)\cap S(g)$, from (5.4), it is easy to see that the poles of $\mathcal {H}_1$ in $\Omega$ which come from the zeros of $f-b_1$ and $f-b_3$ in $\Omega$ are only $S(r, f)+S(r, g)$.

Further, if $z_1$ is a pole of $f$ in $\Omega$ with multiplicity $p$, a pole of $g$ in $\Omega$ with multiplicity $q$, and $b_j(z_1)(b_j(z_1)-1)\neq 0, \infty$ for $j = 1, 2, 3$. W.l.g., assume $p\geq q$, then for $z\rightarrow z_1$, it follows

Thus, $\mathcal {H}_1$ is analytic at $z_1$.

In addition, we can rewrite $\mathcal {H}_1$ as the following form

Thus, if $z_2$ is a zero of $f-b_2$ in $\Omega$ and $b_j(z_1)(b_j(z_1)-1)\neq 0, \infty$ $j = 1, 2, 3$, since $f, g$ share $b_j(z) (j = 1, 2, 3)$ $IM^*$, from the above expression, we know that $\mathcal {H}_1$ is analytic at $z_2$.

Hence, we have $N_0(r, \mathcal {H}_1) = S(r, f)+S(r, g)$, and by combining (5.3), it follows $T_0(r, \mathcal {H}_1) = S(r, f)+S(r, g)$.

Therefore, this completes the proof of Lemma 5.3.

Lemma 5.4. Let $f$ and $g$ be two transcendental or admissible meromorphic functions in $\Omega$, $b_j(z)\in S(f)\cap S(g)\; (j = 1, 2, 3)$. Let $\mathcal {H}_1$ be stated as in Lemma 5.2 and

Then

(i) $\mathcal {H}_1 \equiv -\mathcal {H}_2\equiv -\mathcal {H}_3$;

(ii) $\mathcal {H}_1\equiv0\Longleftrightarrow \widetilde{\mathcal {H}}_1\equiv0$, where

and

Proof. (i) From (5.4), we have

by interchanging between $b_1$ and $b_2$ in (5.5), we can get $\mathcal {H}_1\equiv-\mathcal {H}_2$.

Similarly, by interchanging between $b_2$ and $b_3$ in (5.5), we can get $\mathcal {H}_1\equiv-\mathcal {H}_3$. Thus, (i) is proved.

(ii) Without loss of generality, assume that $b_1 = 0, b_2 = 1, b_3 = b,$ and $b\neq (0, 1, \infty)$. From (i), it follows

Thus, let $F = \frac{1}{f}, G = \frac{1}{g}, \widetilde{b}_1 = 0, \widetilde{b}_2 = 1, \widetilde{b}_3 = \frac{1}{b}$. Similar to the definition of $\mathcal {H}_2$, we have

which implies $\mathcal {H}_1\equiv0\Longleftrightarrow \widetilde{\mathcal {H}}_1\equiv0$ by $b(z)\not\equiv 0$.

Therefore, this completes the proof of Lemma 5.

5.1.   The proof of Theorem 2.2

We will adopt the idea of Yi and Li ^[23], Yao ^[19]. W.l.o.g, we can consider a quasi-Möbius transformation

that is, $\beta_1 = \alpha_1(z), \beta_2 = \alpha_2(z), \beta_3 = 1,$ $\beta_4 = 0$ and $\beta_5(z) = \infty$. Set

If $\mathscr{H}_1\not\equiv0$, from Lemma 5.3, we have

and by Lemma 5.4, it follows

and because

thus it follows

Noting that $\widetilde{E}_\Omega(\alpha_j, l; f) = \widetilde{E}_\Omega(\alpha_j, l; g)$, that is, $\widetilde{N}_0(r, \alpha_j; f|\leq l) = \widetilde{N}_0(r, \alpha_j; g|\leq l)$ for $j = 1, 2, \ldots, 5$, then we have

Similarly, let

then if $\mathscr{H}_j\not\equiv0$ for $j = 2, 3, 4$, by using the same argument as in the above, we have

for $j = 2, 3, 4$. Next, we will prove that

Let

Thus, $\alpha^*_j(z)(j = 1, 2, \ldots, 5)$ are small functions of $F(z)$ and $G(z)$, and $\widetilde{E}_\Omega(\alpha^*_j, l; F) = \widetilde{E}_\Omega(\alpha^*_j, l; G)$ $(j = 1, 2, \ldots, 5)$. Further, by applying Theorem 3.1 and Lemma 5.1 for $F(z), G(z)$, we have

Set

if $\mathscr{H}_5\not \equiv0$, from Lemma 5.3, we have

and by Lemma 5.4, it follows

and since

thus it follows

Noting that $\widetilde{E}_\Omega(\alpha_j, l; f) = \widetilde{E}_\Omega(\alpha_j, l; g)$, that is, $\widetilde{N}_0(r, \alpha_j; f|\leq l) = \widetilde{N}_0(r, \alpha_j; g|\leq l)$ for $j = 1, 2, \ldots, 5$, then from (5.8) we prove (5.7).

By applying Theorem 2.1, we can see that there are at least two of the five $\widetilde{N}_0(r, \alpha_j; f|\leq l)+\widetilde{N}_0(r, \alpha_j; g|\leq l)$ $(j = 1, 2, \ldots, 5)$, w.l.g. assume $j = 2, 3$, such that

and

for $r\geq r_0, r\in I\subseteq[r_0, +\infty)$ and $mes I = +\infty$. Hence, if $\mathscr{H}_2\not \equiv0$, then it follows from (5.6) and (5.9) that

which implies a contradiction with $l\geq 22$. Thus, $\mathscr{H}_2\equiv0$. Similarly, from (5.6) and (5.10) and $l\geq 22$, we have $\mathscr{H}_3\equiv0$.

On the basis of $\mathscr{H}_2\equiv0$ and $\mathscr{H}_3\equiv0$, we have

The five cases will hereinafter be taken into consideration. Let

Case 1. Assume that $\beta_2'(z)\not\equiv0$. If $z_1$ is a zero of $g-\beta_1$ in $\Omega$, and not a zero of $f-\beta_1$ in $\Omega$, and $z_1\not\in J(z)$, $\beta_2'(z_1)\neq 0$, then from (5.11) and (5.12) we have $z_1$ is not a pole of $f-\beta_1$. In fact, if $z_1$ is a zero of $g-\beta_1$ and a pole of $f-\beta_1$ in $\Omega$, and we can rewrite the right sides of (5.11) as $\vartheta_1: = -\frac{g'}{g-1}\frac{f'\beta_1-f\beta_1'}{f-\beta_1}$, then $z_1$ is a pole of $\vartheta_1$ with multiplicity 1, but the left sides of (5.11) is $\vartheta_2: = -\frac{f'}{f-1}\frac{g'\beta_1-g\beta_1'}{g-\beta_1}$, and $z_1$ is a pole of $\vartheta_2$ with multiplicity 2, a contradiction. Similarly, we can get a contradiction from (5.12) when $z_1$ is a zero of $g-\beta_1$ and a pole of $f-\beta_1$ in $\Omega$. Thus, the right sides of (5.11) and (5.12) are analytic at $z_1$. Thus we get $f'(z_1)\neq 0$, $f(z_1)\neq 1$ and $L(f, 1, 0)_{z = z_1} = 0$, $L(f, 0, \beta_2)_{z = z_1} = 0$. Hence it follows $f'(z_1)\neq 0$ and $f(z_1)\beta_2'(z_1)-f'(z_1)\beta_2(z_1) = 0$, that is, $f(z_1)\beta_2'(z_1)-0$, a contradiction with $\beta_2'(z_1)\neq 0$ and $f(z_1)\neq 0$. Thus, we have $f(z_1)-\beta_2(z_1) = 0$.

Hence we get

where $E\subseteq[r_0, +\infty)$, and $mes E < +\infty$. Similarly, we get

which implies that $f, g$ share $\beta_1$ $IM^*$ in $\Omega$.

(ii) Assume that $\beta_2'(z)\equiv0$, that is, $\beta_2(z)$ is a constant, set $\beta_2(z)\equiv \gamma$. Thus, by Lemma 5, (5.12) is equivalent to

Due to $\beta_2\equiv \gamma$, we have $L(f, \beta_2, 0)\equiv\gamma L(f, 1, 0)$, $L(g, \beta_2, 0)\equiv \gamma L(g, 1, 0)$. Substituting them into (5.15), and combining (5.11), we can get

which implies $f\equiv g$.

Case 2. By Lemma 5.4, (5.11) is equivalent to the following equation

(i) $(\beta_1(z)-1)\beta_2'(z)-(\beta_2(z)-1)\beta_1'(z)\not\equiv0$. If $z_2$ is a zero of $f$ in $\Omega$, but not a zero of $g$ in $\Omega$, and $z_2\not\in J(z)$, $(\beta_1(z_2)-1)\beta_2'(z_2)-(\beta_2(z_2)-1)\beta_1'(z_2)\neq0$, then $z_2$ is not a pole of $g$ in $\Omega$. In fact, if $z_2$ is a zero of $f$ and a pole of $g$ in $\Omega$, and we can rewrite the right sides of (5.16) as $\vartheta_3: = \frac{g'}{g}\frac{\beta_1f'-\beta_1'f+\beta_1'-f'}{f-\beta_1}$, then $z_2$ is a pole of $\vartheta_3$ with multiplicity 1, but the left sides of (5.16) is $\vartheta_4: = \frac{f'}{f}\frac{\beta_1g'-\beta_1'g+\beta_1'-g'}{g-\beta_1}$, and by a simply calculation, $z_2$ is a pole of $\vartheta_4$ with multiplicity 2, a contradiction. Similarly, we can get a contradiction from (5.12) when $z_2$ is a zero of $f$ and a pole of $g$ in $\Omega$. Thus, the right sides of (5.12) and (5.16) are analytic at $z_2$. Thus we get $f(z_2)-\beta_1(z_2)\neq0$ and $L(g, 1, \beta_1)_{z = z_2} = 0$, $L(g, \beta_2, \beta_1)_{z = z_2} = 0$. Hence it follows

that is,

as $g(z_2)-\beta_1(z_2)\neq 0$. It means that $(\beta_1(z_2)-1)\beta_2'(z_2)-(\beta_2(z_2)-1)\beta_1'(z_2) = 0$, a contradiction. Hence $z_2$ is a zero of $g(z)$ in $\Omega$, which implies

where $E\subseteq[r_0, +\infty)$, and $mes E < +\infty$. Similarly, we get

Therefore, it means that $f, g$ share 0 $IM^*$ in $\Omega$.

By Lemma 5.3 and from (5.11) and (5.12), it follows

(ii) $(\beta_1(z)-1)\beta_2'(z)-(\beta_2(z)-1)\beta_1'(z)\equiv0$. It follows that $\beta_2-1 = \gamma(\beta_1-1)$ and $\beta_2' = \gamma \beta_1'$, where $\gamma(\neq0)$ a constant. Then, we have

Substituting the above equations into (5.17), and combining (5.16), we get

which implies $f\equiv g$.

Case 3. From (5.11) and (5.15), we have

(i) $\beta_2(z)\beta_1'(z)-\beta_2'(z)\beta_1(z)\not\equiv0$. If $z_3$ is a zero of $g-1$ in $\Omega$, but not a zero of $f-1$ in $\Omega$, and $z_3\not\in J(z)$, $\beta_2(z_3)\beta_1'(z_3)-\beta_2'(z_3)\beta_1(z_3)\neq0$, then $z_3$ is not a pole of $f-1$ in $\Omega$. In fact, if $z_3$ is a zero of $g-1$ and a pole of $f-1$ in $\Omega$, and we can rewrite the right sides of (5.11) as $\vartheta_1: = -\frac{g'}{g-1}\frac{f'\beta_1-f\beta_1'}{f-\beta_1}$, then $z_3$ is a pole of $\vartheta_1$ with multiplicity 2, but the left sides of (5.11) is $\vartheta_2: = -\frac{f'}{f-1}\frac{g'\beta_1-g\beta_1'}{g-\beta_1}$, and by a simply calculation, $z_3$ is a pole of $\vartheta_2$ with multiplicity 1, a contradiction. Similarly, we can get a contradiction from (5.18) when $z_3$ is a zero of $g-1$ and a pole of $f-1$ in $\Omega$. Thus, the left side of (5.11) and the right side of (5.18) are analytic at $z_3$. Hence we get $f(z_3)-\beta_1(z_3)\neq 0$, $f(z_3)-\beta_2(z_3)\neq 0$, but $L(f, \beta_2, 0)_{z = z_3} = 0$ and $L(f, 0, \beta_1)_{z = z_3} = 0$. Then

Since $f, g$ share 0 $IM^*$, and $g(z_3) = 1$, then it follows $f(z_3)\neq 0$. So, we have

which implies $\beta_2(z)\beta_1'(z)-\beta_2'(z)\beta_1(z) = 0$, a contradiction. Hence, it follows that $z_3$ is a zero of $f(z)-1$ in $\Omega$, which implies

where $E\subseteq[r_0, +\infty)$, and $mes E < +\infty$. Similarly, we get

Therefore, it means that $f, g$ share 1 $IM^*$ in $\Omega$.

(ii) $\beta_2(z)\beta_1'(z)-\beta_2'(z)\beta_1(z)\equiv0$. Then it follows $\beta_2 = \gamma \beta_1$ and $\beta_2' = \gamma \beta_1'$, where $\gamma(\neq0)$ a constant. Thus, we have

and

By substituting the above two equivalents into (5.18) and combining (5.11), we have

which implies $f\equiv g$.

Case 4. $\beta_1'(z)\not\equiv0$. If $z_4$ is a zero of $f-\beta_2$ in $\Omega$, but not a zero of $g-\beta_2$ in $\Omega$, and $z_4\not\in J(z)$, $\beta_1'(z_4)\neq 0$, then we have that $z_4$ is not a pole of $g-\beta_2$ in $\Omega$. In fact, if $z_4$ is a zero of $f-\beta_2$ and a pole of $g-\beta_2$ in $\Omega$, and we can rewrite the right sides of (5.15) as $\vartheta_5: = \frac{\beta_2g'-\beta_2'g}{g-\beta_2}\frac{f'\beta_1-f\beta_1'}{f-\beta_1}$, then $z_4$ is a pole of $\vartheta_5$ with multiplicity 1, but the left sides of (5.15) is $\vartheta_6: = \frac{\beta_1g'-\beta_1'g}{g-\beta_1}\frac{f'\beta_2-f\beta_2'}{f-\beta_2}$, and by a simply calculation, $z_4$ is a pole of $\vartheta_6$ with multiplicity 2, a contradiction. Similarly, $z_4$ is a pole of the right side of (5.18) $\vartheta_7: = \frac{f'}{f-1}\frac{\beta_2g'-\beta_2'g}{g-\beta_2}$ with multiplicity 1, and $z_4$ is a pole of the left of (5.18) $\vartheta_8: = \frac{g'}{g-1}\frac{\beta_2f'-\beta_2'f}{f-\beta_2}$ with multiplicity 2. Hence, this is a contradiction. Thus, the right side of (5.15) and (5.18) are analytic at $z_4$. Hence we get $f(z_4)-\beta_1(z_4)\neq 0$, $f(z_4)-1\neq 0$, but $L(g, 0, 1)_{z = z_4} = 0$ and $L(g, 0, \beta_1)_{z = z_4} = 0$. Then $g'(z_4) = 0$, and $g(z_4)\beta_1'(z_4)-g'(z_4)\beta_1(z_4) = 0$, that is, $g(z_4)\beta_1'(z_4) = 0$. Since $\beta_1'(z_4)\neq0$, then it follows $g(z_4) = 0$, which is a contradiction with $f, g$ share 0 $IM^*$ in $\Omega$. Thus, it means that $z_4$ is a zero of $g-\beta_2$ in $\Omega$, which implies

where $E\subseteq[r_0, +\infty)$, and $mes E < +\infty$. Similarly, we get

Therefore, it means that $f, g$ share $\beta_2$ $IM^*$ in $\Omega$.

(ii) $\beta_1'(z)\equiv0$. That is $\beta_1(z)\equiv \delta$, where $\delta(\neq 0)$ a constant. So, it yields $L(f, 0, \beta_1)\equiv \delta L(f, 0, 1)$, and $L(g, 0, \beta_1)\equiv \delta L(g, 0, 1)$. Substituting these into (5.15), and combining (5.18), we can deduce

which implies $f\equiv g$.

Case 5. Let

Then from Lemma 5, we have

and

Further, it follows from $\widetilde{\mathscr{H}}_2\equiv0$ and $\widetilde{\mathscr{H}}_3\equiv0$ that

and

By using the same argument as in Case 2 of Theorem 2.2, we obtain either $F, G$ share 0 $IM^*$ in $\Omega$ or $F\equiv G$. If $F, G$ share 0 $IM^*$ in $\Omega$, it means that $f, g$ share $\infty$ $IM^*$ in $\Omega$. If $F\equiv G$, by a simply calculation, we have $f\equiv g$.

From Cases 1–5, we get that either $f, g$ share $0, 1, \infty, \beta_1(z), \beta_2(z)$ $IM^*$ in $\Omega$, or $f\equiv g$. If $f, g$ share $0, 1, \infty, \beta_1(z), \beta_2(z)$ $IM^*$ in $\Omega$, by using the same method as in ^[22], it follows $f\equiv g$. Hence, we obtain $f\equiv g$.

Thus, this completes the proof of Theorem 2.2.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11961030, 11561033), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001), and the Foundation of Education Department of Jiangxi (GJJ190876, GJJ191042, GJJ190895) of China.

We thank the referees for reading the manuscript very carefully and making a number of valuable and kind comments which improved the presentation.

Conflict of interest

The authors declare that they have no competing interests.