On meromorphic functions for sharing two sets and three sets in m-punctured complex plane

1 Introduction

We firstly assume that readers are familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions such as m(r,f), N(r,f), T(r,f), the first and second main theorems, lemma on the logarithmic derivatives etc. of Nevalinna theory (see Hayman [1], Yang [2] and Yi and Yang [3].

In the past few decades, the uniqueness of meromorphic functions of single connected region attracted many investigations (see [3]) where a number of interesting results were obtained. Around 2000s, Fang, Zheng and Mao investigated the uniqueness of meromorphic functions in the unit disc and some angular domain, and obtained some important results (see [4–9]).

Recently, there were some articles discussing the Nevanlinna theory of meromorphic functions on the annuli (see [10, 11]). In 2004, Korhonen [12] established analogues of Navanlinna’s main theorems including the lemma on the logarithmic derivatives on annuli A:={z:R1≤|z|≤R2} by adopting two parameters R₁,R₂. In 2005 and 2006, Khrystiyanyn and Kondratyuk [13, 14] proposed the Nevanlinna theory for meromorphic functions on annuli A:={z:1R≤|z|≤R} (see also [15]) by adopting one parameter R where 1<R≤+∞ Khrystiyanyn and Kondratyuk [13, 14], and Kondratyuk and Laine [15] obtained a series of results of value distribution and uniqueness of meromorphic functions on annuliA:={z:1R≤|z|≤R}where1<R≤+∞ including the first and second main theorems, lemma on the logarithmic derivatives on annuli, also including five-values theorem of Nevanlinna on annulus. In2010, Fernández [16] further investigated the value distribution of meromorphic functions on annulus and gave some extension of some results about meromorphic functions in the plane with finitely many poles. At about the same time, Cao [17, 18] investigated the uniqueness of meromorphic functions on annuli sharing some values and some sets, and obtained a number of results which is an improvement of the five-values theorem of Nevanlinna on annulus given by [15]. In 2012, Cao and Deng [19] investigated the uniqueness of meromorphic functions that share three or two finite sets on annulus, and obtained that there exist three sets S₁,S₂,S₃ with ♯ S₁ = ♯S₂ = 1 and ♯ S₃ = 5, such that any two admissible meromorphic functions f and g must be identical if f, g share S₁,S₂,S₃CM on annuliA In the same year, Xu and Xuan [20] further investigated the problem of meromorphic functions sharing four values on annulus, and gave a theorem which is also an improvement of the five-values theorem of Nevanlinna on annuli given by [15].

As we all know, annulus is a double connected region, can be regarded as a special multiply connected region. Thus, it is natural to ask: what results can we get when meromorphic functions f, g share some values or finite sets on the multiply connected region? However, there is no paper discussing uniqueness for meromorphic functions in the multiply connected region. The main purpose of this article is to investigate the uniqueness of meromorphic functions in a special multiply connected region—m-punctured complex plane.

The structure of this paper is as follows. In Section 2, we introduce the basic notations andfundamental theorems of meromorphic functions m-punctured complex plane. Section 3 is devoted to study the uniqueness of meromorphic functions that share three finite sets I M in m-punctured complex planes. Section 4 is devoted to give the uniqueness theorem for meromorphic functions sharing two finite sets I M in m-punctured complex planes.

2 Nevanlinna theory in m-punctured complex planes

We call thatΩ=C∖⋃j=1m{cj} is an m-punctured complex plane, wherecj∈C,j∈{1,2,…,m},m∈N+ are distinct points. The annulus is regarded as a special m-punctured plane if m = 1 which is studied by [13, 14]. The main purpose of this article is to study meromorphic functions of those m-punctured planes for which m ≥ 2.

Denoted=12min{|ck−cj|:j≠k}andr0=1d+max{|cj|:j∈{1,2,…,m}} Then

D¯1/r0(cj)∩D¯1/r0(ck)=∅forj≠kandD¯1/r0(cj)⊂Dr0(0)forj∈{1,2,…,m} whereDδ(c)={z:|z−c|<δ}andD¯δ(c)={z:|z−c|≤δ} For an arbitrary r ≥r0, we define

Thus, it follows that Ω_r ⊃ Ω_r0 for _r0<r ≤ + ∞. It is easy to see that Ω_r is m + 1 connected region. In 2007, Hanyak and Kondratyuk [21] proposed the Nevanlinna value distribution theory for meromorphic functions in m-punctured complex planes and proved a number of theorems which are analog of those results on the whole plane ℂ

Let f be a meromorphic function in an m-punctured plane Ω, we use n₀(r,f) to denote the counting function of its poles in Ω¯r,r0≤r<+∞ and

and we also define

wherelog+⁡x=max{log⁡x,O} then we call that

is the Nevanlinna characteristic of f.

3 Meromorphic functions share two sets IM

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 inC^andΩ⊆C

Define

where fa(z)=f(z)−aifa∈Candf∞(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 ΩifEΩ(S,f)=EΩ(S,g) we say f and g share the set S IM (ignoring multiplicities) in Ω if E¯Ω(S,f)=E¯Ω(S,g) In particular, when S={a}, where a∈C^ we say f and g share the value aC M in Ω if EΩ(S,f)=EΩ(S,g) and we say f and g share the value aIMinΩifE¯Ω(S,f)=E¯Ω(S,g)

Then P(w) is a uniqueness polynomial in a broad sense if and only if

Proof of Theorem 3.2. Set F=P1(f)andG=P1(g).SinceE¯Ω(Sj,f)=E¯Ω(Sj,g), then we have that fg share 0,1I M in Ω and F′=P1′(f)=f4(f−1)4f,′G′=g4(g−1)4g′ From Lemma 3.6, we have T0(r,F)=9T0(r,f)+S(r,f),T0(r,G)=9T0(r,g)+S(r,g)andS(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 λ(>12)andasetI⊂[r0,+∞)(mesI=+∞) such that

Set U=F′F−G′G, from Theorem 2.4 we have m0(r,U)=S(r,F)+S(r,G)=S(r,f)+S(r,g) Suppose thatU≢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

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

Hence,

From (5)-(7), it follows that for r_∄ r < +∞

By adding N~0(r,1F)+N~0(r,1G) into both sides of (74), and from E¯Ω(f,S1)=E¯Ω(f,S1),forr0<_r<+∞ we have

Thus, we can deduce by applying Theorem 2.5 and (67) that

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

where K a non-zero constant. From Lemma 3.5 we have

The three following subcases will be considered.

Subcase 1.1. Suppose that K=1. Thus, if follows from (10) that F≡G that is,

From the form of P₁(w), we can see that there exist nine distinct complex constants α_j(j=1,2;…},9) such that

Moreover, we have P1′(w)=w4(w−1)4has mutually distinct two zeros 0;1 with multiplicities 4,4, respectively, and satisfying 4 × 4=16 > 8 = 4 + 4. Thus, P₁(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=ζ1 , where ζ1=19−12+157−23+15+1.Obviously,ζ1≠0,1. Then from (10) we have F≡ζ1G, that is,

It follows that 0,1 is a Picard exceptional value of f, g in Ω. In fact, if there exists z₀∈ Ω such that f(z_o)=1, since E¯Ω(S1,f)=E¯Ω(S1,g)then g(z₀)=1. Thus from (13), we have that ζ1−1=ζ12−1,which implies ζ₁=0 or ζ₁=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 ζ1P1(w)−1,obviously,βv≠0,1. It is easy to find that P₁(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

Since 0 is a Picard exceptional of f in Ω, by applying Theorem 2.4 for above equality, it follows that

which is a contradiction with (11).

Subcase 1.3. Suppose that K≠1andK≠ζ1. From (10), we have

It is easy to see that 0 is a Picard exceptional value of f, g in Ω. In fact, if there exists z₀∈ Ω such that f(z₀)=0, since f, g share OIM in Ω, then F(z₀)=G(z₀)=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 P₁(w)-K in Ω, obviously, β_v}≠ 0,1. Similar to Subcase 1.2, we have

Since 0 is a Picard exceptional of g in Ω, by applying Theorem 2.4 for above equality, it follows that

which is a contradiction with (11).

Case 2. Suppose that there exist a constant κ(12<_κ<712)andasetI⊂[r0,+∞)(mesI=+∞) such that

as r→+∞,r∈I Set

From [21, Lemma 6] we have m0(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

where N~0∗(r,1f′) is the reduced counting function of those zeros of f^′ in Ω which are not the zeros of f(f-1) and N~0∗(r,1g′) is similarly defined. From Lemma 3.4, we have N¯01)E(r,1F)=N~01)E(r,1G)<_N0(r,1H) where N~01)E(r,1F) 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

Let V=f′g′f(f−1)g(g−1), by Lemma 3.6 we have m0(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

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

Since EΩ(S1,f)=EΩ(S1,g) from (20), (21) and Theorem 2.2, we have

Noting that N~0[2(r,1F)<_N0∗(r,1f′)andN~0[2(r,1G)<_N0∗(r,1g′) , then from (19)-(22) we have for r₀ ≤ r < + ∈

where N~0[2(r,1F) is the reduced counting function of those zeros of f with multiply ≥ 2, and N~0[2(r,1G) is similarly defined.

Similarly, for r₀ ≤ r < + ∈ we have

as r₀ ≤ r < +∈. By applying Theorem 2.4 and from (23) and (24), we have

which is a contradiction with κ<712are transcendental in Ω.

Thus, H ≡ 0, i.e.,

By integration, we have from (22) that 1F=AG+Bwhere A,B are constants which are not equal to zero at the same time.

Suppose that B≠0.Thus,1F=A+BGG From Lemma 3.5, we have T0(r,f)+S(r,f)=T0(r,g)+S(r,g)forr0<_r<+∞. Moreover, it follows from Theorem 2.4 that

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.

4 Meromorphic functions in m-punctured plane share three sets IM

In this section, we will investigate the uniqueness of meromorphic functions in m-punctured plane sharing three sets with finite elements IM. The main result of this chapter is showed as follows.