# Borel Directions and Uniqueness of Meromorphic Functions

Authors: Keyu Zhang; HongYan Xu; Hongxun Yi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the relationship between Borel directions and uniqueness of meromorphic functions and obtain some results of meromorphic functions sharing four distinct values IM and one set in an angular domain containing a Borel line. Our result is an improvement of a recent theorem given by Long and Wu (2012).


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## Body

## 1. Introduction and Main Results

We useℂ to denote the open complex plane, ℂ^(=ℂ⋃‍{∞}) to denote the extended complex plane, and Ω (⊂ℂ) to denote an angular domain. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used in [1, 2]. In addition, the order of meromorphic function f is defined by
(1)ρ=ρ(f)=limsupr→∞logT(r,f)logr.LetS be a set of distinct elements in ℂ^ and Ω:={z:α≤argz≤β}⊆ℂ. Define
(2)E(S,Ω,f)=⋃a∈S‍{z∈Ω∣fa(z)=0,counting multiplicities},E¯(S,Ω,f)=⋃a∈S‍{z∈Ω∣fa(z)=0,ignoring multiplicities},
where fa(z)=f(z)-a if a∈ℂ and f∞(z)=1/f(z).Letf and g be two nonconstant meromorphic functions in ℂ. If E(S,Ω,f)=E(S,Ω,g),  then we say f and g share the set S CM (counting multiplicities) in Ω. If E¯(S,Ω,f)=E¯(S,Ω,g),  then we say f and g share the set S IM (ignoring multiplicities) in Ω. In particular, when S={a}, a∈ℂ^, we say f and g share the value a CM in Ω if E(S,Ω,f)=E(S,Ω,g), and we say f and g share the value a IM in Ω if E¯(S,Ω,f)=E¯(S,Ω,g). When Ω=ℂ, we give the simple notation as before, E(S,f),E¯(S,f), and so on (see [3]).Nevanlinna (see [4]) proved the following well-known theorems.Theorem 1 (see [4]).
Iff and g are two nonconstant meromorphic functions that share five distinct values a1,a2,a3,a4,a5 IM in ℂ, then f(z)≡g(z).Theorem 2 (see [4]).
Iff and g are two distinct nonconstant meromorphic functions that share four distinct values a1,a2,a3,a4 CM in ℂ, then f is a Möbius transformation of g, two of the shared values, say a1 and a2, are Picard values, and the cross ratio (a1,a2,a3,a4)=-1.After their very work, many investigations studied the uniqueness of meromorphic functions with shared values in the whole complex plane (see [5]). Zheng studied the uniqueness problem under the condition that five values and four values are shared in some angular domain in ℂ around 2003 (see [6, 7]). It is an interesting topic to investigate the uniqueness with shared values in the angular domain; see [3, 6–12]. The basic notations and definitions of meromorphic functions in an angular domain will be introduced as follows (see [1, 6, 7]).Letf be a meromorphic function on the angular domain Ω(α,β)={z:α≤argz≤β} and 0<β-α≤2π. Define
(3)Aα,β(r,f)=ωπ∫1r‍(1tω-tωr2ω)×{log+|f(teiα)|+log+|f(teiβ)|}dtt,Bα,β(r,f)=2ωπrω∫αβ‍log+|f(reiθ)|sinω(θ-α)dθ,Cα,β(r,f)=2∑1<|bμ|<r‍(1|bμ|ω-|bμ|ωr2ω)sinω(θμ-α),Sα,β(r,f)=Aα,β(r,f)+Bα,β(r,f)+Cα,β(r,f),
where ω=π/(β-α) and bμ=|bμ|eiθμ(μ=1,2,…) are the poles of f on Ω(α,β) counted according to their multiplicities. Sα,β(r,f) is called the Nevanlinna’s angular characteristic, and Cα,β(r,f) is called the angular counting function of the poles of f on Ω(α,β), and C¯α,β(r,f) is the reduced function of Cα,β(r,f). Similarly, when a≠∞, we will use the notations Aα,β(r,1/(f-a)), Bα,β(r,1/(f-a)), Cα,β(r,1/(f-a)), Sα,β(r,1/(f-a)), and so on.It is well known that angular distribution is an interesting topic of value distribution of meromorphic function in complex analysis, and Borel directions played an important role in the topic of angular distribution (see [13–24]). Valiron [16] proved that every meromorphic function of finite order ρ>0 has at least one Borel direction of order ρ. Chuang [25] investigated the existence of Borel direction of meromorphic function of infinite order. To state the Chuang’s results, we will introduce the definition as follows.Definition 3 (see [25]).
Letf be a meromorphic function of infinite order, ρ(r) is a real function satisfying the following conditions:(i)
ρ(r) is continuous, nondecreasing for r≥r0 and ρ(r)→∞ as r→∞;(ii)
(4)limr→∞logU(R)logU(r)=1,R=r+rlogU(r),whereU(r)=rρ(r)(r≥r0);(iii)
(5)limsupr→∞logT(r,f)logU(r)=1.Then ρ(r) is called infinite order of meromorphic function f. This definition is given by Chuang [25].Letρ(r) be infinite order of meromorphic function f; we will denote by M(ρ(r)) the set of meromorphic function g satisfying limsupr→∞(logT(r,g)/ρ(r)logr)≤1; that is,
(6)M(ρ(r)):={g:limsupr→∞logT(r,g)ρ(r)logr≤1}.Letα<β, β-α<2π, r>0, and Ω(α,β,r):={z:α≤argz≤β,0<|z|≤r}. The definition of Borel direction of meromorphic functions f of infinite order ρ(r) is  given as follows.Definition 4 (see [25]).
Letf be meromorphic functions of infinite order ρ(r); if for any ɛ(0<ɛ<π), the equality
(7)limsupr→∞logn(Ω(θ-ɛ,θ+ɛ,r),f=a)ρ(r)logr=1
holds for any complex number a∈ℂ^, at most except two exceptions, where n(Ω(θ-ɛ,θ+ɛ,r),f=a) is the counting function of  zeros of the function f-a in the angular domain Ω(θ-ɛ,θ+ɛ), counting multiplicities, then the ray argz=θ is called a Borel direction of ρ(r) order of meromorphic function f.Remark 5.
Chuang [25] proved that every meromorphic function f  of infinite order ρ(r) has at least one Borel direction of infinite order ρ(r).In 2012, Long and Wu [26] investigated the problem concerning Borel direction and shared value of meromorphic functions and obtained the following theorems.Theorem 6 (see [26, Theorem 1.1]).
Letf be a meromorphic function of infinite order ρ(r), and let g∈M(ρ(r)), argz=θ(0≤θ<2π) be one Borel direction of ρ(r) order of meromorphic function f; let ai∈ℂ^(i=1,2,3,4,5) be five distinct complex numbers. If f and g share ai(i=1,2,3,4,5) IM in the angular domain Ω(θ-ɛ,θ+ɛ) for any ɛ(0<ɛ<π), then f≡g.Theorem 7 (see [26, Theorem 1.2]).
Letf be a meromorphic function of infinite order ρ(r), and let g∈M(ρ(r)), argz=θ(0≤θ<2π) be one Borel direction of ρ(r) order of meromorphic function f; let ai∈ℂ^(i=1,2,3,4) be four distinct complex numbers. If f and g share ai(i=1,2,3,4) CM in the angular domain Ω(θ-ɛ,θ+ɛ) for any ɛ(0<ɛ<π), then f is a Möbius transformation of g.
Thus, a question arises naturally:  could the nature of sharing the valuesαi be further relaxed in Theorems 6 and 7?In this paper, we will deal with the above question and obtain the following result which is an improvement of Theorem6.Theorem 8.
Letf be a meromorphic function of infinite order ρ(r), and let g∈M(ρ(r)), argz=θ(0≤θ<2π) be one Borel direction of ρ(r) order of meromorphic function f; we assume that f and g share four distinct values aj(j=1,2,3,4) IM in Ω(θ-ɛ,θ+ɛ), and E¯(S,Ω(θ-ɛ,θ+ɛ),f)⊂E¯(S,Ω(θ-ɛ,θ+ɛ),g), for any ɛ(0<ɛ<π), where S={b1,…,bm}, m≥1 and b1,…,bm∈ℂ^∖{a1,a2,a3,a4}. Then f and g share all values CM; thus, it follows that either f≡g or f is a Möbius transformation of g. Furthermore, if the number of the values in S is odd, then f≡g.Remark 9.
The special casem=1 of this theorem immediately yields Theorem 6. In fact, when m=1, set S={a5}. If f, g share a5 IM, which implies E¯(S,Ω(θ-ɛ,θ+ɛ),f)⊂E¯(S,Ω(θ-ɛ,θ+ɛ),g), then by Theorem 8, we can get f≡g.


## 2. Some Lemmas

To prove our result, we require the following lemmas.Lemma 10 (see [27]).
Letf be a nonconstant meromorphic function on Ω(α,β). Then for arbitrary complex number a, we have
(8)Sα,β(r,1f-a)=Sα,β(r,f)+ɛ(r,a),
where ɛ(r,a)=O(1) as r→∞.Lemma 11 (see [7, 28]).
Suppose thatf is a nonconstant meromorphic function in one angular domain Ω(α,β) with 0<β-α≤2π; then, for arbitrary q distinct aj∈ℂ^(1≤j≤q), we have
(9)(q-2)Sα,β(r,f)≤∑j=1q‍C¯α,β(r,1f-aj)+Rα,β(r,f),
where the term C¯α,β(r,1/(f-aj)) will be replaced by C¯α,β(r,f) when some aj=∞ and
(10)Rα,β(r,f)=Aα,β(r,f′f)+Bα,β(r,f′f)+∑j=1q‍{Aα,β(r,f′f-aj)+Bα,β(r,f′f-aj)}+O(1).Lemma 12 (see [27, Page 138]).
Letf be a nonconstant meromorphic function in the whole complex plane ℂ. Given one angular domain on Ω(α,β). Then, for any 1≤r<R, we have
(11)Aα,β(r,f′f)≤K{(Rr)ω∫1R‍log+T(r,f)t1+ωdt+log+rR-r+logRr+1(Rr)ω∫1R‍log+T(r,f)t1+ω(Rr)ω∫1R‍},(12)Bα,β(r,f′f)≤4ωrωm(r,f′f),
where ω=π/(β-α) and K is a positive constant not depending on r and R.Remark 13.
Nevanlinna conjectured that(13)Aα,β(r,f′f)+Bα,β(r,f′f)=o(Sα,β(r,f))
when r tends to +∞ outside an exceptional set of finite linear measure, and he proved that Aα,β(r,f′/f)+Bα,β(r,f′/f)=O(1) when the function f is meromorphic in ℂ and has finite order. In 1975, Goldberg [28] constructed a counter example to show that (13) is not valid.Remark 14.
From Lemmas11 and 12, we can get the following conclusion:
(14)Rα,β(r,f)={O(1),fis offinite order,O(logU(r)),r∉E,fis ofinfinite order,
where Rα,β(r,f) is stated as in (10), U(r)=rρ(r),ρ(r) is infinite order of meromorphic function f, and E is a set of finite linear measure.Remark 15.
From the definition ofAα,β(r,f), Bα,β(r,f), Cα,β(r,f), Sα,β(r,f), and Lemmas 10–12, we can see that the properties of Cα,β(r,f),(A+B)α,β(r,f), and Sα,β(r,f) are the same as for the more familiar quantities N(r,f), m(r,f), and T(r,f), respectively.Lemma 16 (see [29]).
Letf be meromorphic function of infinite order ρ(r). Then the ray argz=θ is one Borel direction of ρ(r) order of meromorphic function f if and only if f satisfies the equality
(15)limsupr→∞logSθ-ϵ,θ+ϵ(r,f)ρ(r)logr=1,
for any ϵ(0<ϵ<π/2).By using the same argument as in [8, Lemma 1] and [5], we can get the lemma below easily.Lemma 17.
Suppose thatf is a nonconstant meromorphic function with infinite order ρ(r), and the ray argz=θ is one Borel direction of ρ(r) order of meromorphic function f. Let P(f)=a0fp+a1fp-1+⋯+ap(a0≠0) be a polynomial of f with degree p, where the coefficients aj(j=0,1,…,p) are constants, and let bj(j=1,2,…,q) be q(q≥p+1) distinct finite complex numbers. Then for any ɛ(0<ɛ<π/2),
(16)(A+B)θ-ɛ,θ+ɛ(r,P(f)·f′(f-b1)(f-b2)⋯(f-bq))=Rθ-ɛ,θ+ɛ(r,f).Lemma 18.
Suppose thatf is a nonconstant meromorphic function with infinite order ρ(r), and the ray argz=θ is one Borel direction of ρ(r) order of meromorphic function f. We assume that for any for any ɛ(0<ɛ<π/2), f and g share four distinct values aj(j=1,2,3,4)  IM in angular domain Ω(θ-ɛ,θ+ɛ) and f≢g. Let
(17)Ψ=f′g′(f-g)2×((f-a1)(f-a2)(f-a3)(f-a4)×(g-a1)(g-a2)×(g-a3)(g-a4))-1.
Then, Sθ-ɛ,θ+ɛ(r,Ψ)=Rθ-ɛ,θ+ɛ(r,f)+Rθ-ɛ,θ+ɛ(r,g).Proof.
Since the rayargz=θ is one Borel direction of meromorphic function f  of ρ(r) order, thus, for any ɛ(0<ɛ<π/2), we can assume that z0∈Ω(θ-ɛ,θ+ɛ) and f(z0)=a1 (or a2,a3,a4) with multiplicity p and g(z0)=a1 (or a2,a3,a4) with multiplicity q. From (17), we can get
(18)Ψ(z)=O((z-z0)2min(p,q)-2).
Hence, Ψ is analytic in Ω(θ-ɛ,θ+ɛ). By Lemmas 12 and 17, we have
(19)Sθ-ɛ,θ+ɛ(r,Ψ)=(A+B)θ-ɛ,θ+ɛ(r,Ψ)≤(A+B)θ-ɛ,θ+ɛ×(r,f′(f-a2)(f-a3)(f-a4))+(A+B)θ-ɛ,θ+ɛ×(r,f′(f-a1)(f-a2)(f-a3))+(A+B)θ-ɛ,θ+ɛ×(r,f′(f-a1)(f-a2)(f-a3)(f-a4))+(A+B)θ-ɛ,θ+ɛ×(r,f′P1(f)(f-a1)(f-a2)(f-a3)(f-a4))+(A+B)θ-ɛ,θ+ɛ×(r,g′(g-a2)(g-a3)(g-a4))+(A+B)θ-ɛ,θ+ɛ×(r,g′(g-a1)(g-a2)(g-a3))+(A+B)θ-ɛ,θ+ɛ×(r,g′(g-a1)(g-a2)(g-a3)(g-a4))+(A+B)θ-ɛ,θ+ɛ×(r,g′P2(g)(g-a1)(g-a2)(g-a3)(g-a4))+O(1)=Rθ-ɛ,θ+ɛ(r,f)+Rθ-ɛ,θ+ɛ(r,g),
where P1(f) is a polynomial of degree no more than 2 in f and P2(g) is a polynomial of degree no more than 2 in g.
Thus, we complete the proof of this lemma.


## 3. Proof of Theorem8

Proof.
Sincef is a meromorphic function of infinite order ρ(r) and argz=θ(0≤θ<2π) is one Borel direction of ρ(r) order of the meromorphic function f, by Lemma 16, we can get for any ɛ(0<ɛ<π)(20)limsupr→∞logSθ-ɛ,θ+ɛ(r,f)ρ(r)logr=1.
And since g∈M(ρ(r)), we have
(21)limsupr→∞logSθ-ɛ,θ+ɛ(r,g)ρ(r)logr≤1.
Set R(r)=O(ρ(r)logr) as r→∞,r∉E where E is a set of finite linear measure; then R(r)=o(Sθ-ɛ,θ+ɛ(r,f)) as r→∞, r∉E.
Sincef and g share four distinct values aj(j=1,2,3,4) IM in Ω(θ-ɛ,θ+ɛ), suppose that f≢g and none of the aj(j=1,2,3,4) is ∞. By Lemma 11, we have
(22)2Sθ-ɛ,θ+ɛ(r,f)≤∑j=14‍C¯θ-ɛ,θ+ɛ(r,1f-aj)+Rθ-ɛ,θ+ɛ(r,f)≤Cθ-ɛ,θ+ɛ(r,1f-g)+Rθ-ɛ,θ+ɛ(r,f)≤Sθ-ɛ,θ+ɛ(r,f)+Sθ-ɛ,θ+ɛ(r,g)+Rθ-ɛ,θ+ɛ(r,f),
and by interchanging f and g, we can get that
(23)Sθ-ɛ,θ+ɛ(r,f)=Sθ-ɛ,θ+ɛ(r,g)+Rθ-ɛ,θ+ɛ(r,f),Sθ-ɛ,θ+ɛ(r,g)=Sθ-ɛ,θ+ɛ(r,f)+Rθ-ɛ,θ+ɛ(r,g).
Thus, it follows Rθ-ɛ,θ+ɛ(r,g)=Rθ-ɛ,θ+ɛ(r,f)=R(r) and
(24)∑j=14‍C¯θ-ɛ,θ+ɛ(r,1f-aj)=2Sθ-ɛ,θ+ɛ(r,f)+R(r).
LetΨ be the function expressed in Lemma 18. Then, Ψ≢0. By Lemma 11 and (24), for any bj∈S(j=1,2,…,m), we have
(25)3Sθ-ɛ,θ+ɛ(r,f)≤∑j=14‍C¯θ-ɛ,θ+ɛ(r,1f-aj)+C¯θ-ɛ,θ+ɛ(r,1f-bj)+Rθ-ɛ,θ+ɛ(r,f)≤2Sθ-ɛ,θ+ɛ(r,f)+C¯θ-ɛ,θ+ɛ(r,1f-bj)+R(r);
that is,
(26)C¯θ-ɛ,θ+ɛ(r,1f-bj)=Sθ-ɛ,θ+ɛ(r,f)+R(r).
Similarly, we have
(27)C¯θ-ɛ,θ+ɛ(r,1g-bj)=Sθ-ɛ,θ+ɛ(r,g)+R(r).
From (26) and (27), it follows
(28)(A+B)θ-ɛ,θ+ɛ(r,1f-bj)=R(r),(A+B)θ-ɛ,θ+ɛ(r,1g-bj)=R(r),
for any bj∈S(j=1,2,…,m).
Set(29)Ψ1:=(g-b1)⋯(g-bm)(f-b1)⋯(f-bm)·(g′(f-g)(g-a1)⋯(g-a4))m,Ψ2:=(f-b1)⋯(f-bm)(g-b1)⋯(g-bm)·(f′(f-g)(f-a1)⋯(f-a4))m.
By Lemma 17 and (28), we can get that
(30)(A+B)θ-ɛ,θ+ɛ(r,1f-bj·g′(f-g)(g-bj)(g-a1)⋯(g-a4))=R(r),(A+B)θ-ɛ,θ+ɛ(r,1g-bj·f′(f-g)(f-bj)(f-a1)⋯(f-a4))=R(r).
Thus, it follows that (A+B)θ-ɛ,θ+ɛ(r,Ψj)=R(r), j=1,2. From (28), we see that “almost all” of poles and bj-points of f and g in the angular domain Ω(θ-ɛ,θ+ɛ) are simple. Since f, g share the four distinct values aj, j=1,2,3,4 in the angular domain Ω(θ-ɛ,θ+ɛ) and E¯(S,Ω(θ-ɛ,θ+ɛ),f)⊂E¯(S,Ω(θ-ɛ,θ+ɛ),g), we can easily get that Cθ-ɛ,θ+ɛ(r,Ψ1)=R(r). Therefore, we have
(31)Sθ-ɛ,θ+ɛ(r,Ψ1)=R(r).
SinceΨ1Ψ2≡Ψm, we can have
(32)Sθ-ɛ,θ+ɛ(r,Ψ2)=R(r).
LetΞΩpq(aj) be the set of those aj-points of f and g in the angular domain Ω(θ-ɛ,θ+ɛ) in which the multiplicities of f and g at these points are p and q, respectively. For any z0∈ΞΩpq(a1), by simple computation, we have
(33)Ψ1(z0)=(q·f′(z0)-g′(z0)(a1-a2)(a1-a3)(a1-a4))m,Ψ2(z0)=(p·f′(z0)-g′(z0)(a1-a2)(a1-a3)(a1-a4))m.
Hence,
(34)1qmΨ1(z0)-1pmΨ2(z0)=0.
Similarly, we can see that (34) holds for any z0∈ΞΩpq(aj),j=2,3,4.
Now, two cases will be considered below.
Case  1. Suppose that Ψpq:=(1/qm)Ψ1-(1/pm)Ψ2≢0, for all positive integers p, q.
First, we useCθ-ɛ,θ+ɛpq(r,1/(f-aj)) to denote the counting function of f in Ω(θ-ɛ,θ+ɛ) with respect to the set ΞΩpq(aj), and we also use C¯θ-ɛ,θ+ɛpq(r,1/(f-aj)) to denote the corresponding reduced counting function. Thus, we have
(35)Cθ-ɛ,θ+ɛ(r,1f-aj)=∑p,q=1∞‍Cθ-ɛ,θ+ɛpq(r,1f-aj),C¯θ-ɛ,θ+ɛ(r,1f-aj)=∑p,q=1∞‍C¯θ-ɛ,θ+ɛpq(r,1f-aj).
From the above two equations and (28)–(32), we can see that Sθ-ɛ,θ+ɛ(r,Ψpq)=R(r). And by (34) each zero of f-aj is a zero of Ψpq, so with the help of Lemma 10 and Ψpq≢0, we can get
(36)C¯θ-ɛ,θ+ɛpq(r,1f-aj)≤C¯θ-ɛ,θ+ɛpq(r,1Ψpq)≤Sθ-ɛ,θ+ɛ(r,1Ψpq)≤Sθ-ɛ,θ+ɛ(r,Ψpq)+O(1)=R(r),
for some p,q. Since Sθ-ɛ,θ+ɛ(r,f)=Sθ-ɛ,θ+ɛ(r,g)+R(r), it follows
(37)C¯θ-ɛ,θ+ɛ(r,1f-aj)=∑max(p,q)≥5‍C¯θ-ɛ,θ+ɛpq(r,1f-aj)+R(r,f)≤15∑max(p,q)≥5‍Cθ-ɛ,θ+ɛpq(r,1f-aj)+15∑max(p,q)≥5‍Cθ-ɛ,θ+ɛpq(r,1g-aj)+R(r)≤15(Cθ-ɛ,θ+ɛ(r,1f-aj)+Cθ-ɛ,θ+ɛ(r,1g-aj))+R(r)≤25Sθ-ɛ,θ+ɛ(r,f)+R(r),j=1,2,3,4.
From the above inequality and (24), we can get
(38)2Sθ-ɛ,θ+ɛ(r,f)≤85Sθ-ɛ,θ+ɛ(r,f)+R(r).
Since f is of infinite order ρ(r) and R(r)=O(ρ(r)logr), we can get a contradiction to (38).
Case  2. Suppose that Ψpq:=(1/qm)Ψ1-(1/pm)Ψ2≡0, for some positive integers p, q. From the definitions of Ψ1 and Ψ2, we have
(39)(pq)m·(g-b1)2⋯(g-bm)2(f-b1)2⋯(f-bm)2≡(f′(g-a1)⋯(g-a4)g′(f-a1)⋯(f-a4))m.
Next, we take the following two subcases into consideration.
Subcase  2.1. Suppose that p≠q. Without loss of generality, we may assume that p<q. For some two positive integers p1 and q1, if z1∈ΞΩp1q1(aj) for some j∈{1,2,3,4}, then (39) implies that p/q=p1/q1. Hence, q1>p1≥1, and q1≥2 which means that any aj-points (j=1,2,3,4) of g in Ω(θ-ɛ,θ+ɛ) are multiple. By Lemma 11 and (24), we can get
(40)2Sθ-ɛ,θ+ɛ(r,g)=∑j=14‍C¯θ-ɛ,θ+ɛ(r,1g-aj)+R(r)≤12∑j=14‍Cθ-ɛ,θ+ɛ(r,1g-aj)+R(r)≤2Sθ-ɛ,θ+ɛ(r,g)+R(r);
it follows that
(41)Sθ-ɛ,θ+ɛ(r,g)=Cθ-ɛ,θ+ɛ(r,1g-aj)+R(r),(42)Cθ-ɛ,θ+ɛ(r,1g-aj)=2C¯θ-ɛ,θ+ɛ(r,1g-aj)+R(r).
From (41) and (42), we can see that “almost all” of aj-points of g have multiplicity 2, and “almost all” of aj-points of f are simple in Ω(θ-ɛ,θ+ɛ). Without loss of generality, we may assume that f and g attain the values a3 and a4 in Ω(θ-ɛ,θ+ɛ). Set
(43)Φ1≔2f′(f-a4)(f-a1)(f-a2)(f-a3)-g′(g-a4)(g-a1)(g-a2)(g-a3),Φ2≔2f′(f-a3)(f-a1)(f-a2)(f-a4)-g′(g-a3)(g-a1)(g-a2)(g-a4).
SinceΦi(i=1,2) is analytic at the poles of f and of g and also at those common aj-points of f and g which have multiplicity 1 with respect to f and multiplicity 2 with respect to g, by Lemma 17, we have Sθ-ɛ,θ+ɛ(r,Φi)=R(r),i=1,2. If Φi≢0, then Cθ-ɛ,θ+ɛ(r,1/(g-a4))≤Cθ-ɛ,θ+ɛ(r,1/ϕ1)=R(r), which contradicts to (41). Then, Φ1≡0. Similarly, we have Φ2≡0. Therefore, from the definitions of Φ1 and Φ2, we have
(44)(f-a4f-a3)2≡(g-a4g-a3)2.
Sincef≢g, from (44), we have
(45)f-a4f-a3≡-g-a4g-a3,
which implies that f and g share a3,a4 CM in Ω(θ-ɛ,θ+ɛ). Since f and g assume the value a3, there exist positive integers p1,q1 such that ΞΩp1q1(a3)≠∅. From the considerations above we get q1>p1, contradicting the fact that f and g share a3 CM.
Subcase  2.2. Suppose that p=q.
In this subcase, (39) becomes
(46)(g-b1)2⋯(g-bm)2(f-b1)2⋯(f-bm)2≡(f′(g-a1)⋯(g-a4)g′(f-a1)⋯(f-a4))m.
which implies that f and g share the four values aj(j=1,2,3,4) CM in Ω(θ-ɛ,θ+ɛ). Then by applying Theorem 7, g is a Möbius transformation of f. Furthermore, two of the four values, say a1, a2, are Picard exceptional values of f and g in Ω(θ-ɛ,θ+ɛ). Set
(47)Λ1≔f′(f-a4)(f-a1)(f-a2)(f-a3)-g′(g-a4)(g-a1)(g-a2)(g-a3),Λ2≔f′(f-a3)(f-a1)(f-a2)(f-a4)-g′(g-a3)(g-a1)(g-a2)(g-a4).
Similar to the discussion in Subcase 2.1 for Λ1, Λ2, we can get
(48)f-a3f-a4≡-g-a3g-a4.
We define the Möbius transformations T, H, and J by
(49)T(ω):=w-a3w-a4,H(w):=-w,J:=T-1∘H∘T.
Then, we have
(50)T∘f=-T∘g;hence,g=J∘f.
Obviously, a3 and a4 are the fixed points of J. Therefore, there exist no fixed points of J in the set S. Let some b∈S be given. Then, in vies of b≠a1,a2, there exists a z0∈ℂ such that b=f(z0), and from E¯(S,Ω(θ-ɛ,θ+ɛ),f)⊆E¯(S,Ω(θ-ɛ,θ+ɛ),g) we obtain
(51)J(b)=J(f(z0))=g(z0)∈S.
So S is invariant under J. Furthermore, we have J∘J=I where I denotes the identical transformation. Hence, S must contain an even number of values. Thus, the proof of Theorem 8 is completed.



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*Source: 793810-2013-08-21.xml*