In this section, we will discuss the uniqueness of meromorphic functions in *m*-punctured planes that shared two sets with finite elements *IM*. Some basic notations of uniqueness of meromorphic functions would be introduced as follows.
Let *S* be a set of distinct elements in$\hat{\mathbb{C}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\Omega}\subseteq \mathbb{C}$

Define$${E}_{\mathrm{\Omega}}(S,f)={\displaystyle \bigcup _{a\in S}-a\{z\in \mathrm{\Omega}|{f}_{a}(z)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{counting}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{multiplicities};}$$
$${\overline{E}}_{\mathrm{\Omega}}(S,f)={\displaystyle \bigcup _{a\in S}\{z\in \mathrm{\Omega}|{f}_{a}(z)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{ignoring}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{multiplicities};}$$
where ${f}_{a}(z)=f(z)-a\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a\in \mathbb{C}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{f}_{\mathrm{\infty}}(z)=1/f(z)$

Let *f* and *g* be two non-constant meromorphic functions in ℂ, we say *f* and *g* share the set *S CM* (counting multiplicities) in $\mathrm{\Omega}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{E}_{\mathrm{\Omega}}(S,f)={E}_{\mathrm{\Omega}}(S,g)$
we say *f* and *g* share the set *S IM* (ignoring multiplicities) in *Ω* if ${\overline{E}}_{\mathrm{\Omega}}(S,f)={\overline{E}}_{\mathrm{\Omega}}(S,g)$
In particular, when *S*={a}, where $a\in \hat{\mathbb{C}}$
we say *f* and *g* share the value *a* *C M* in *Ω* if ${E}_{\mathrm{\Omega}}(S,f)={E}_{\mathrm{\Omega}}(S,g)$
and we say *f* and *g* share the value *a* $IM\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\Omega}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\overline{E}}_{\mathrm{\Omega}}(S,f)={\overline{E}}_{\mathrm{\Omega}}(S,g)$

*Let* *f* *be a nonconstant meromorphic function in* *m*-*punctured plane* *Ω*. *The function* *f* *is called transcendental in* *m*-*punctured plane* *Ω* *provided that*$$\underset{r\to +\mathrm{\infty}}{lim\u2006sup}\frac{{T}_{0}(r,f)}{\mathrm{log}r}=+\mathrm{\infty},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{r}_{0}\le r<+\mathrm{\infty}.$$
Now, we will show my first main theorem of this article as follows.

*Let* *f* *and* *g* *be two transcendental meromorphic functions in* *Ω*, *and let* *S*_{1}={0,1},*S*_{2} = *w : P*_{1}(w)=0}, *where*$${P}_{1}(w)=\frac{{w}^{9}}{9}-\frac{4{w}^{8}}{8}+\frac{15{w}^{7}}{7}-\frac{4{w}^{6}}{6}+\frac{{w}^{5}}{5}+1.$$
$If{\overline{E}}_{\mathrm{\Omega}}({S}_{j,}f)={\overline{E}}_{\mathrm{\Omega}}({S}_{j,}g)(j=1,2)$

*There exist two sets* *S*_{1},*S*_{2} *with*#*S*_{1} = 2 *and*#*S*_{2} = 9, *such that any two transcendental meromorphicfunctions* *f* *and* *g* *must be identical* if
${\overline{E}}_{\mathrm{\Omega}}({S}_{j;}f)={\overline{E}}_{\mathrm{\Omega}}({S}_{j;}g)(j=1,2)$ *where* #S *is to denote the cardinality of a set S*.

To prove this theorem, we require some lemmas as follows.

*Let* *f, g* *be two non-constant meromorphicfunctions in* *m*-*puncturedplane* *Ω*, *and let z*_{0} be a common pole of *f, g* *in* *Ω* *with multiply 1, then z*_{0} is a zero of
$\frac{{f}^{\u2033}}{{f}^{\prime}}-\frac{{g}^{\u2033}}{g}$ *in Ω* *with multiply k* ≥ 1.

From the assumptions of this lemma, we can set
$$f(z)=\frac{\phi (z)}{z-{z}_{0}},g(z)=\frac{\psi (z)}{z-{z}_{0}},$$

where φ(*z*), ψ(*z*) are analytic in *Ω* and φ(*z*_{0})ψ(*z*)_{0} ≠ 0, then
$${f}^{\prime}(z)=\frac{\phi \mathrm{\prime}(z)(z-{z}_{0})-\phi (z)}{(z-{z}_{0}{)}^{2}};{f}^{\u2033}(z)=\frac{\phi \mathrm{\prime}\mathrm{\prime}(z)(z-{z}_{0}{)}^{2}-2\phi \mathrm{\prime}(z)(z-{z}_{0})-2\phi (z)}{(z-{z}_{0}{)}^{3}}.$$

It follows that
$$\frac{{f}^{\u2033}}{{f}^{\prime}}=(z-{z}_{0})\frac{\phi \mathrm{\prime}\mathrm{\prime}(z)}{\phi \mathrm{\prime}(z)(z-{z}_{0})-\phi (z)}-\frac{2}{z-{z}_{0}}.$$

Similarly, we have
$$\frac{{g}^{\u2033}}{{g}^{\prime}}=(z-{z}_{0})\frac{{\psi}^{\u2033}(z)}{{\psi}^{\prime}(z)(z-{z}_{0})-\psi (z)}-\frac{2}{z-z0}.$$

Thus, it follows that
$$\frac{{f}^{\u2033}}{{f}^{\prime}}-\frac{{g}^{\u2033}}{{g}^{\prime}}=(z-{z}_{0})\zeta (z),$$

where ξ(*z*) is analytic at *z*_{0} in *Ω*. Therefore, we prove the conclusion of this lemma.

By a similar discussion as in [22], we can obtain a stand and Valiron-Mohon’ko type theorem in *Ω* as follows.

*Let* *f* *be a nonconstant meromorphic function in* *m-punctured plane* *Ω*, *and let*
$$R(f)=\sum _{k=0}^{n}{a}_{k}{f}^{k}/\sum _{j=0}^{m}{b}_{j}{f}^{j}$$

*be an irreducible rational function in* *f* *with coecients* {*a*_{k}} *and* {*b*_{j}}, *where a*_{n} ≠ 0 *and b*_{m} ≠ 0. *Then*
$${T}_{0}(r,R(f))=d{T}_{0}(r,f)+S(r,f),$$

*where* d = max{*n, m*}.

*Suppose that* *f* *is a transcendental meromorphic function in* *m*-*punctured plane* *Ω*. *Let*
$Q(f)={a}_{0}{f}^{p}+{a}_{1}{f}^{p-1}+\cdots +{a}_{p}({a}_{0}\ne 0)$ *be apolynomial of* *f* *with degree p*, *where the coecients a*_{j} (*j* *= 0, 1, …, p) are constants, and let* *b*_{j} (j = 1, 2, …, q) be q (q ≥ p+1) distinct finite complex numbers. Then
$${m}_{0}\left(r,\frac{Q(f)\cdot {f}^{\prime}}{(f-{b}_{1})(f-{b}_{2})\dots (f-{b}_{q})}\right)=S(r,f),$$

*where S(r,f)* *is stated as in Theorem 2.3*.

Since $$\frac{Q(f)\cdot {f}^{\prime}}{(f-{b}_{1})(f-{b}_{2})\dots (f-{b}_{q})}=\sum _{j=1}^{q}\frac{{\varphi}_{j}}{f-{b}_{j}},$$

where ϕ_{j} are non-zero constants. Then, it follows from Theorem 2.4 that
$$\begin{array}{ll}{m}_{0}(r,\frac{Q(f)\cdot {f}^{\prime}}{(f-{b}_{1})(f-{b}_{2})\cdots (f-{b}_{q})})& ={m}_{0}(r,\sum _{j=1}^{q}\frac{{\varphi}_{j}{f}^{\prime}}{f-{b}_{j}})\\ & \underset{\_}{<}\sum _{j=1}^{q}{m}_{0}(r,\frac{{\varphi}_{j}{f}^{\prime}}{f-{b}_{j}})+O(1)\\ & \underset{\_}{<}S(r,f)\end{array}.$$

Thus, this completes the proof of Lemma 3.6.

([23]). *We also call P(w) a uniqueness polynomial in a broad sense if P(f)*= *P(g) implies f* = *g for any nonconstant meromorphic functions f,g*.

(see [23]). *Let S* = {*a*_{1},*a*_{2},…,*a*_{q}}, *a*_{1},*a*_{2},…,*a*_{q} *be q distinct complex constants, P(w) be a monic polynomial of the form P(w)*=(*w*-*a*_{1})(*w*-*a*_{2})…(*w*-*a*_{q}). *If P* '(*w*) *has mutually distinct k zeros e*_{1},*e*_{2},… *e*_{k} *with multiplicities q*_{1},q_{2},…,q_{k} *respectively and satisfies*
$$P({e}_{\ell})\ne P({e}_{m}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\underset{\_}{<}\ell <m\underset{\_}{<}k.$$

*Then P(w) is a uniqueness polynomial in a broad sense if and only if*
$$\sum _{1\le \ell <m\le k}q\ell {q}_{m}>\sum _{\ell =1}^{k}q\ell .$$

*Proof of Theorem 3.2*. Set
$F={P}_{1}(f)\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}G={P}_{1}(g).\phantom{\rule{thinmathspace}{0ex}}Since\phantom{\rule{thinmathspace}{0ex}}{\overline{E}}_{\mathrm{\Omega}}({S}_{j,}f)={\overline{E}}_{\mathrm{\Omega}}({S}_{j,}g),$
then we have that *f**g* share 0,1*I M* in *Ω* and
${F}^{\prime}={P}_{1}^{\prime}(f)={f}^{4}(f-1{)}^{4}f{,}^{\prime}{G}^{\prime}={g}^{4}(g-1{)}^{4}{g}^{\prime}$
From Lemma 3.6, we have
${T}_{0}(r,F)=9{T}_{0}(r,f)+S(r,f),{T}_{0}(r,G)=9{T}_{0}(r,g)+S(r,g)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}S(r,F)=S(r,f),S(r,G)=S(r,g).$
Next, the following two cases will be discussed.

Case 1: Suppose that there exist a constant
$\lambda (>\frac{1}{2})\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}set\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}I\subset [{r}_{0},+\mathrm{\infty})(\mathit{m}\mathit{e}\mathit{s}\mathit{I}=+\mathrm{\infty})$
such that
$${\stackrel{~}{N}}_{0}(r,\frac{1}{f})+{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})\underset{\_}{>}\lambda ({T}_{0}(r,f)+{T}_{0}(r,g))+S(r,f)+S(r,g),(r\to +\mathrm{\infty},r\in I).$$
Set
$U={\displaystyle \frac{{F}^{\prime}}{F}-\frac{{G}^{\prime}}{G},}$
from Theorem 2.4 we have
${m}_{0}(r,U)=S(r,F)+S(r,G)=S(r,f)+S(r,g)$
Suppose that$U\not\equiv 0$ since *f*,*g* share 0,1 *I M* in Ω, we can see that the common zeros of *f, g* are the zero of *U* in *Ω*, and the common zeros of *f*-1,*g*-1 are also the zero of *U* in *Ω*. Thus, we have
$$4{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+4{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})\underset{\_}{<}{N}_{0}(r,\frac{1}{U}).$$
On the other hand, it is easy to see that the pole of *U* in *Ω* may occur at the poles of *f, g* or the zeros of *f, g* in *Ω*. Then it follows that
$${N}_{0}(r,U){\displaystyle \underset{\_}{<}{\stackrel{~}{N}}_{0}(r,f)+{\stackrel{~}{N}}_{0}(r,g)+{\stackrel{~}{N}}_{0}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}(r,\frac{1}{G})}$$
Hence,
$${T}_{0}(r,U){\displaystyle \underset{\_}{<}{\stackrel{~}{N}}_{0}(r,f)+{\stackrel{~}{N}}_{0}(r,g)+{\stackrel{~}{N}}_{0}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}(r,\frac{1}{G})+S(r,f)+S(r,g)}$$
From (5)-(7), it follows that for *r*_{∄} r < +∞
$$\begin{array}{ll}4{\stackrel{~}{N}}_{0}(r,\frac{1}{f})& +4{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})\underset{\_}{<}{N}_{0}(r,\frac{1}{U})\underset{\_}{<}{T}_{0}(r,\frac{1}{U})+S(r,f)\\ & \underset{\_}{<}{\stackrel{~}{N}}_{0}(r,f)+{\stackrel{~}{N}}_{0}(r,g)+{\stackrel{~}{N}}_{0}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}(r,\frac{1}{G})+S(r,f)+S(r,g).\end{array}$$

By adding
${\stackrel{~}{N}}_{0}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}(r,\frac{1}{G})$
into both sides of (74), and from
${\overline{E}}_{\mathrm{\Omega}}(f,{S}_{1})={\overline{E}}_{\mathrm{\Omega}}(f,{S}_{1}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}$
we have
$$\begin{array}{ll}{\stackrel{~}{N}}_{0}(r,\frac{1}{F})& +{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})+{\stackrel{~}{N}}_{0}(r,\frac{1}{G})+{\stackrel{~}{N}}_{0}(r,\frac{1}{g})+{\stackrel{~}{N}}_{0}(r,\frac{1}{g-1})+\\ & +2{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+2{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})\\ & \underset{\_}{<}2{\stackrel{~}{N}}_{0}(r,\frac{1}{F})+2{\stackrel{~}{N}}_{0}(r,\frac{1}{G})+{\stackrel{~}{N}}_{0}(r,f)+{\stackrel{~}{N}}_{0}(r,g)+S(r,f)+S(r,g).\end{array}$$

Thus, we can deduce by applying Theorem 2.5 and (67) that
$$9\{{T}_{0}(r,f)+{T}_{0}(r,g)\}+2\lambda ({T}_{0}(r,f)+{T}_{0}(r,g))+S(r,f)+S(r,g)\underset{\_}{<}10\{{T}_{0}(r,f)+{T}_{0}(r,g)\}+S(r,f)+S(r,g),r\to +\mathrm{\infty},r\in I.$$(9)

Since λ > 0 and *f, g* are admissible functions in *Ω*, we can get a contradiction. Thus, it follows that *U* ≡ 0, by integration, we have
$$F\equiv KG$$(10)

where *K* a non-zero constant. From Lemma 3.5 we have
$${T}_{0}(r,f)={T}_{0}(r,g)+S(r,g),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}.$$(11)

The three following subcases will be considered.

**Subcase 1.1**. Suppose that *K*=1. Thus, if follows from (10) that
$F\equiv G$
that is,
$${P}_{1}(f)\equiv {P}_{1}(g)$$(12)

From the form of *P*_{1}(*w*), we can see that there exist nine distinct complex constants α_{j}(*j*=1,2;…},9) such that
$${P}_{1}(w)=(w-{\alpha}_{1})(w-{\alpha}_{2})\cdots (w-{\alpha}_{9}).$$

Moreover, we have
${P}_{1}^{\prime}(w)={w}^{4}(w-1{)}^{4}$has mutually distinct two zeros 0;1 with multiplicities 4,4, respectively, and satisfying 4 × 4=16 > 8 = 4 + 4. Thus, *P*_{1}(*w*) is a uniqueness polynomial in a broad sense. From Lemma 3.8, we can get that *f* ≡ *g*.

**Subcase 1.2**. Suppose that
$K={\zeta}_{1}$
, where
${\zeta}_{1}=\frac{1}{9}-\frac{1}{2}+\frac{15}{7}-\frac{2}{3}+\frac{1}{5}+1.\phantom{\rule{thinmathspace}{0ex}}\text{Obviously},\phantom{\rule{thinmathspace}{0ex}}{\zeta}_{1}\ne 0,1.$
Then from (10) we have
$F\equiv {\zeta}_{1}G,$
that is,
$$F-1\equiv {\zeta}_{1}G-1.$$(13)

It follows that 0,1 is a Picard exceptional value of *f, g* in Ω. In fact, if there exists *z*_{0}∈ Ω such that *f*(*z*_{o})=1, since
${\overline{E}}_{\mathrm{\Omega}}({S}_{1,}f)={\overline{E}}_{\mathrm{\Omega}}({S}_{1,}g)$then *g*(*z*_{0})=1. Thus from (13), we have that
${\zeta}_{1}-1={\zeta}_{1}^{2}-1,$which implies ζ_{1}=0 or ζ_{1}=1, a contradition. Similarly, we can get that 0 is a Picard exceptional value of *f, g* in Ω.

Let β_{v}(*v*=1,2,…,9) be nine distinct roots of equation ${\zeta}_{1}{P}_{1}(w)-1,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{obviously},\phantom{\rule{thinmathspace}{0ex}}{\beta}_{v}\ne 0,1.$
It is easy to find that *P*_{1}(*w*)-1 have one root 0 with order 5 and four distinct roots, say α_{t}(*t*= 1,2,3,4) . Thus, we can deduce from (11) that
$$\sum _{v=1}^{9}{\stackrel{~}{N}}_{0}(r,\frac{1}{g-{\beta}_{v}})={\stackrel{~}{N}}_{0}(r,\frac{1}{f})+\sum {}_{t=1}^{4}{\stackrel{~}{N}}_{0}(r,\frac{1}{f-{\alpha}_{t}}),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}.$$

Since 0 is a Picard exceptional of *f* in Ω, by applying Theorem 2.4 for above equality, it follows that
$$7{T}_{0}(r,g)+S(r,g)\underset{\_}{<}4{T}_{0}(r,f)+S(r,f),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty},$$

which is a contradiction with (11).

**Subcase 1.3**. Suppose that
$K\ne 1\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}K\ne {\zeta}_{1}.$
From (10), we have
$$F-K\equiv K(G-1)$$

It is easy to see that 0 is a Picard exceptional value of *f*, *g* in Ω. In fact, if there exists *z*_{0}∈ Ω such that *f*(*z*_{0})=0, since *f, g* share O*IM* in Ω, then *F*(*z*_{0})=*G*(*z*_{0})=1. Thus, we can deduce from (14) that 1-*K* ≡ 0, a contradiction. Similarly, we can prove that 0 is a Picard exceptional value of *g* in Ω.

Let γ *v*(*v* = 1, 2, …, 9) be nine distinct roots of *P*_{1}(*w*)-*K* in Ω, obviously, β_{v}}≠ 0,1. Similar to Subcase 1.2, we have
$$\sum _{v=1}^{9}{\stackrel{~}{N}}_{0}(r,\frac{1}{f-{\gamma}_{v}})={\stackrel{~}{N}}_{0}(r,\frac{1}{g})+\sum _{t=1}^{4}{\stackrel{~}{N}}_{0}(r,\frac{1}{g-{\alpha}_{t}}),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}$$(15)

Since 0 is a Picard exceptional of *g* in Ω, by applying Theorem 2.4 for above equality, it follows that
$$7{T}_{0}(r,f)+S(r,g)\underset{\_}{<}4{T}_{0}(r,g)+S(r,f),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty},$$

which is a contradiction with (11).

**Case 2**. Suppose that there exist a constant
$\kappa (\frac{1}{2}\underset{\_}{<}\kappa <\frac{7}{12})\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\text{a}\phantom{\rule{thinmathspace}{0ex}}\text{set}\phantom{\rule{thinmathspace}{0ex}}I\subset [{r}_{0,}+\mathrm{\infty})(\mathit{m}\mathit{e}\mathit{s}\mathit{I}=+\mathrm{\infty})$
such that
$${\stackrel{~}{N}}_{0}(r,\frac{1}{f})+{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})\underset{\_}{<}\kappa ({T}_{0}(r,f)+{T}_{0}(r,g))+S(r,f)+S(r,g),$$(16)

as
$r\to +\mathrm{\infty},r\in I$
Set
$$H={\displaystyle \frac{(\frac{1}{F}{)}^{\u2033}}{(\frac{1}{F}{)}^{\prime}}-\frac{(\frac{1}{G}{)}^{\u2033}}{(\frac{1}{G}{)}^{\prime}}=(\frac{{F}^{\u2033}}{{F}^{\prime}}-\frac{2{F}^{\prime}}{F})-(\frac{{G}^{\u2033}}{{G}^{\prime}}-\frac{2{G}^{\prime}}{G}).}$$(17)

From [21, Lemma 6] we have
${m}_{0}(r,H)=S(r,F)+S(r,G)=S(r,f)+S(r,g).$

Suppose that *H*≢ 0, we know that the pole of *H* in *Ω* may occur at the zeros of *F*^{′} *G*^{′} in Ω and the poles of *f, g* in Ω. Then we have
$${N}_{0}(r,H){\displaystyle \underset{\_}{<}{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})+{\stackrel{~}{N}}_{0}(r,f)+{\stackrel{~}{N}}_{0}(r,g)+}+{\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{f}^{\mathrm{\prime}}})+{\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{g}^{\mathrm{\prime}}}),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}.$$(18)

where
${\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{f}^{\prime}})$
is the reduced counting function of those zeros of *f*^{′} in Ω which are not the zeros of *f*(*f*-1) and
${\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{g}^{\mathrm{\prime}}})$
is similarly defined. From Lemma 3.4, we have
${\overline{N}}_{0}^{1)E}(r,\frac{1}{F})={\stackrel{~}{N}}_{0}^{1)E}(r,\frac{1}{G})\underset{\_}{<}{N}_{0}(r,\frac{1}{H})$
where
${\stackrel{~}{N}}_{0}^{1)E}(r,\frac{1}{F})$
is the counting function of those common zeros of *f, g* with multiply 1 in Ω. Then it follows from Theorem 2.2 and (18) that
$$\begin{array}{l}{\stackrel{~}{N}}_{0}^{1)E}(r,\frac{1}{F})\underset{\_}{<}{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})+{\stackrel{~}{N}}_{0}(r,f)+{\stackrel{~}{N}}_{0}(r,g)+\\ \phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}+{\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{f}^{\mathrm{\prime}}})+{\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{g}^{\mathrm{\prime}}})+S(r,f)+S(r,g),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}.\end{array}$$(19)

Let
$V={\displaystyle \frac{{f}^{\prime}{g}^{\prime}}{f(f-1)g(g-1)},}$
by Lemma 3.6 we have
${m}_{0}(r,V)=S(r,f)+S(r,g).$
Noting that the zeros of *f*^{′} in Ω which are not the zeros of *f*,*f*-1 in Ω may be the zeros of *V* in Ω, and the zeros of *g*^{′} in Ω which are not the zeros of *g*,*g* -l in Ω may also be the zeros of *V* in Ω then
$${\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{f}^{\mathrm{\prime}}})+{\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{g}^{\mathrm{\prime}}})\underset{\_}{<}{N}_{0}(r,\frac{1}{V}),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}.$$(20)

On the other hand, the poles of *V* in Ω can occur at the zeros of *f*,*f*-1,*g* or *g*-1 in Ω. It follows that
$${N}_{0}(r,V){\displaystyle \underset{\_}{<}{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})+{\stackrel{~}{N}}_{0}(r,\frac{1}{g})+{\stackrel{~}{N}}_{0}(r,\frac{1}{g-1}),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}.}$$(21)

Since
${E}_{\mathrm{\Omega}}({S}_{1,}f)={E}_{\mathrm{\Omega}}({S}_{1,}g)$
from (20), (21) and Theorem 2.2, we have
$${\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{f}^{\mathrm{\prime}}})+{\stackrel{~}{N}}_{0}^{\ast}(r,\frac{1}{{g}^{\mathrm{\prime}}})\underset{\_}{<}2{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+2{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})+S(r,f)+S(r,g),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}.$$(22)

Noting that
${\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{F})\underset{\_}{<}{N}_{0}^{\ast}(r,\frac{1}{{f}^{\prime}})\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{G})\underset{\_}{<}{N}_{0}^{\ast}(r,\frac{1}{{g}^{\mathrm{\prime}}})$
, then from (19)-(22) we have for *r*_{0} ≤ *r* < + ∈
$$\begin{array}{ll}{\stackrel{~}{N}}_{0}(r,\frac{1}{F})& ={\stackrel{~}{N}}_{0}^{1)E}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{G})\underset{\_}{<}{N}_{0}(r,\frac{1}{H})+{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{G})\\ & \underset{\_}{<}{T}_{0}(r,H)+{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{G})+O(1)\\ & \underset{\_}{<}{N}_{0}(r,H)+{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{G})+S(r,f)\\ & \underset{\_}{<}{\stackrel{~}{N}}_{0}(r,f)+{\stackrel{~}{N}}_{0}(r,g)+5{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+5{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})+S(r,f)+S(r,g),\end{array}$$(23)

where
${\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{F})$
is the reduced counting function of those zeros of *f* with multiply ≥ 2, and
${\stackrel{~}{N}}_{0}^{[2}(r,\frac{1}{G})$ is similarly defined.

Similarly, for *r*_{0} ≤ *r* < + ∈ we have
$${\stackrel{~}{N}}_{0}(r,\frac{1}{G})\underset{\_}{<}{\stackrel{~}{N}}_{0}(r,f)+{\stackrel{~}{N}}_{0}(r,g)+5{\stackrel{~}{N}}_{0}(r,\frac{1}{g})+5{\stackrel{~}{N}}_{0}(r,\frac{1}{g-1})+S(r,f)+S(r,g),$$(24)

as *r*_{0} ≤ *r* < +∈. By applying Theorem 2.4 and from (23) and (24), we have
$$\begin{array}{ll}9\{{T}_{0}(r,f)& +{T}_{0}(r,g)\}\underset{\_}{<}{\stackrel{~}{N}}_{0}(r,\frac{1}{F})+{\stackrel{~}{N}}_{0}(r,\frac{1}{G})+2\{{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})\}\\ & +S(r,f)+S(r,g)\\ & \underset{\_}{<}2{N}_{0}(r,f)+2{N}_{0}(r,g)+12\{{\stackrel{~}{N}}_{0}(r,\frac{1}{f})+{\stackrel{~}{N}}_{0}(r,\frac{1}{f-1})\}+\\ & +S(r,f)+S(r,g)\\ & \underset{\_}{<}(2+12\kappa )\{{T}_{0}(r,f)+{T}_{0}(r,g)\}+S(r,f)+S(r,g),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty},\end{array}l$$25

which is a contradiction with
$\kappa <\frac{7}{12}$are transcendental in Ω.

Thus, H ≡ 0, i.e.,
$$\frac{{F}^{\u2033}}{{F}^{\prime}}-\frac{2{F}^{\prime}}{F}\equiv \frac{{G}^{\u2033}}{{G}^{\prime}}-\frac{2{G}^{\prime}}{G}.$$(26)

By integration, we have from (22) that
$\frac{1}{F}=\frac{A}{G}+B$where *A,B* are constants which are not equal to zero at the same time.

Suppose that
$B\ne 0.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}Thus,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{1}{F}=\frac{A+BG}{G}}$
From Lemma 3.5, we have
${T}_{0}(r,f)+S(r,f)={T}_{0}(r,g)+S(r,g)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{r}_{0}\underset{\_}{<}r<+\mathrm{\infty}$. Moreover, it follows from Theorem 2.4 that
$${\stackrel{~}{N}}_{0}(r,f)={\stackrel{~}{N}}_{0}(r,F)={\stackrel{~}{N}}_{0}(r,\frac{1}{G-\frac{A}{B}})\underset{\_}{>}3{T}_{0}(r,g)+S(r,g),{r}_{0}\underset{\_}{<}r<+\mathrm{\infty},$$

which is a contradiction with *f, g* are transcendental in Ω.

Suppose that *B* ≡ 0. Then *G*=*AF* where *A* is a non-zero constant. Similarly to the same argument as in Case 1, we can get that *A* ≡ 1. By Lemma 3.8, we can get *f* ≡ *g* easily.

From Case 1 and Case 2, we can get the conclusion of Theorem 3.2.

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