# The Filling Discs Dealing with Multiple Values of an Algebroid Function in the Unit Disc

Authors: Zu-Xing Xuan; Nan Wu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

In this paper, by using the potential theory we prove the existence of filling discs dealing with multiple values of an algebroid function of finite order defined in the unit disc.


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## Body

## 1. Introduction and Main Result

The value distribution theory of meromorphic functions due to Hayman (see [1] for standard references) was extended to the corresponding theory of algebroid functions by Selbreg [2], Ullrich [3], and Valiron [4] around 1930. The filling discs of an algebroid function are an important part of the value distribution theory. For an algebroid function defined on z-plane, the existence of its filling discs was proved by Sun [5] in 1995. In 1997, for the algebroid functions of infinite order and zero order, Gao [6] obtained the the corresponding results. The existence of the sequence of filling discs of algebroid functions dealing with multiple values, of finite or infinite order, was first proved by Gao [7, 8]. The existence of filling discs in the strong Borel direction of algebroid function with finite order was proved by Huo and Kong in [9]. Compared with the case of ℂ, it is interesting to investigate the algebroid functions defined in the unit disc, and there are some essential differences between these two cases. Recently, the first author [10] has investigated this problem and confirmed the existence of filling discs for this case. In this note, we will continue the work of Xuan [10] by considering the case dealing with multiple values and get more precise results.Letw=w(z)(z∈Δ) be the ν-valued algebroid function defined by irreducible equation(1.1)Aν(z)wν+Aν-1(z)wν-1+⋯+A0(z)=0,
where Aν(z),…,A0(z) are entire functions without any common zeros. The single-valued domain of definition of w(z) is a ν-valued covering of the z-plane, a Riemann surface, denoted by R̃z. A point in R̃z, whose projection in the z-plane is z, is denoted by z̃. The part of R̃z, which covers the disc {z:|z|<r}, is denoted by |z̃|<r.Denote(1.2)S(r,w)=1π∫∫|z̃|≤r[|w′(z)|1+|w(z)|2]2dω.S(r,w)is called the mean covering number of |z̃|≤r into w-sphere under the mapping w=w(z). And S(r,w) is conformal invariant. Let n(r,a) be the number of zeros of w(z)-a, counted according to their multiplicities in |z̃|≤r.n¯l)(E,w=α) denotes the number of zeros with multiplicity ≤l of w(z)=α in E, each zero being counted only once.Let(1.3)N(r,a)=1ν∫0rn(t,a)-n(0,a)tdt+n(0,a)νlog⁡⁡r,m(r,a)=12πν∫|z̃|=r∑j=1νlog⁡+|1wj(reiθ)-a|dθ,z=reiθ,
where |z̃|=r is the boundary of |z̃|≤r. The characteristic function of w(z) is defined by(1.4)T(r,w)=1ν∫0rS(t,w)tdt.
In view of [4], we have(1.5)T(r,w)=m(r,w)+N(r,∞)+O(1).
The order of algebroid function w(z) is defined by(1.6)ρ=limsup⁡r→1-log⁡⁡T(r,w)log⁡⁡(1/(1-r)).In this paper we assume that0<ρ<+∞,V is the w-sphere, and C is a constant which can stand for different constant. Let n(r,R̃z) be the number of the branch points of R̃z in |z̃|≤r, counted with the order of branch. Write(1.7)N(r,R̃z)=1ν∫0rn(t,R̃z)-n(0,R̃z)tdt+n(0,R̃z)νlog⁡⁡r.Valiron is the first one to introduce the concept of a proximate orderρ(1/(1-r)) for a meromorphic function w(z) with finite positive order and U(1/(1-r))=(1/(1-r))ρ(1/(1-r)) is called type function of w(z) or T(r,w) such that ρ(1/(1-r)) is nondecreasing, piecewise continuous, and differentiable, and(1.8)lim⁡r→1-ρ(11-r)=ρ,lim⁡r→1-U(k/(1-r))U(1/(1-r))=kρ(kisanygivenpositiveconstant),lim⁡r→1-T(r,w)U(1/(1-r))=1,lim⁡r→1-(1/(1-r))ρ-εU(1/(1-r))=0,0<ε<ρ.For an algebroid functionw(z) of finite positive order, we can apply the same method to get its type function U(1/(1-r)).Our main result is the following.Theorem 1.1.
Suppose thatw(z) is the  ν-valued algebroid function of finite order  ρ in  |z|<1 defined by (1.1) and  l(≥ 2ν+1) is an integer, then there exists a sequence of discs
(1.9)Γn:{|z-zn|<rnσn},n=1,2,…,
where
(1.10)zn=rneiθn,lim⁡n→∞rn=1,σn>0,lim⁡n→∞σn=0.
Such that for each α(1.11)n¯l)(Γn∩▵,w=α)≥1(1-rn)ρ+1-εn,
except for those complex numbers contained in the union of 2ν spherical discs each with radius (1-rn)ρ/11, where limn→+∞εn=0,Δ={z:|z|<1}.
The discs with the above property are called filling discs dealing with multiple values.Remark 1.2.
In [10], the result says that n¯(Γn∩▵,w=α)≥1/(1-rn)ρ-εn. Theorem 1.1 is really the improvement of [10].Remark 1.3.
The existence of filling discs in Borel radius of meromorphic functions was proved by Kong [11]. In view of our theorem, we can get the similar results of [11] (when ν=1). But we must point out that the structure and definition of filling discs between this paper and [11] are different. There are also some papers relevant to the singular points of algebroid functions in the unit disc (see [12–14]).


## 2. Two Lemmas

Lemma 2.1 (see [15] or [16]).
Suppose thatw(z) is the  ν-valued algebroid function in  {z:|z|<R} defined by (1.1),  l(≥2ν+1) is an integer and  a1,a2,a3,…,aq(q≥3) are distinct points given arbitrarily in  w-sphere, and the spherical distance of any two points is no smaller than  δ∈(0,1/2), then for any  r∈(0,R), one has
(2.1)(q-2-2l)S(r,w)≤∑j=1qn¯l)(R,aj)+l+1ln(R,R̃z)+CR(R-r)δ10.Combining the potential theory with Lemma2.1, one proves Lemma 2.2, which is crucial to the theorem.Lemma 2.2.
Suppose thatw(z) is the  ν-valued algebroid function of finite order  ρ satisfying  0<ρ<+∞ in  |z|<1 defined by (1.1) and  l(≥2ν+1) is an integer. For any  ε∈(0,ρ),0<R<1, there exists  a0∈(1/2,1), such that for any  a∈(a0,1), put
(2.2)rn=1-an,m=[2π1-a],θq=2π(q+1)m,Ωpq={1-ap≤|z|<1-ap+2}∩{|arg⁡z-θq|≤2πm}(p=1,2,3,…;q=0,1,2,…,m-1),
where [x] stands for the inter part of x.
Then, amongp,q, there exists at least one pair p0,q0, such that 1-ap0>R, and in Ωp0q0,
(2.3)n¯l)(Ωp0q0,w=α)≥1aρ+1-ε,
except for those complex numbers contained in the union of 2ν spherical discs each with radius δ=ap0ρ/11.Proof.
Suppose the conclusion is false. Then there exists a sequence{ai}i=1∞(0<ai<1), where limi→∞ai=1. For any a∈{ai}, any p>P=log(1-R)/loga and q∈{0,1,2,…,m-1}, there exist 2ν+1 complex numbers which satisfy that the spherical distance of any two of those points is no smaller than δ=apρ/40. Denote
(2.4){αj=αj(p,q)}j=12ν+1.
For any p,q mentioned above, we have
(2.5)n¯l)(Ωpq,w=αj)<1ap(ρ+1-ε).
For anyr>R, let T=[log(1-r)/loga], then we have 1-aT≤r<1-aT+1.
For any given positive integersN and M, set
(2.6)b=a1/M∈(0,1),γpt=1-bMp+t,t=0,1,2,…,M-1,Lpt={γpt≤|z|<γp,t+1},θqj=2πqm+2πjNm,▵qj={z:|z|<1-aT,θqj≤arg⁡z<θq,j+1}.
Then
(2.7){1-a≤|z|<1-aT}=⋃t=0M-1⋃p=1T-1Lpt,{|z|<1-aT}=⋃j=0N-1⋃q=0m-1▵qj.
Thus there exists t0,j0 which are related to T. We can assume t0=0,j0=0, such that
(2.8)∑p=1T-1n(Lp0,R̃z)≤1Mn(1-aT,R̃z),∑q=0m-1n(▵q0,R̃z)≤1Nn(1-aT,R̃z).
Set
(2.9)Ωpq0={1-bMp+bMp+12≤|z|<1-bMp+M+bMp+M+12}∩{θq0+θq12≤arg⁡z<θq+1,0+θq+1,12},Ω¯pq={1-bMp≤|z|<1-bMp+M+1}∩{θq0≤arg⁡z<θq+1,1}.
Then we have
(2.10)Ωpq0⊂Ω¯pq⊂Ωpq.
Since {Ω¯pq}p,q covers ⋃p=1T-1Lp0 and ⋃q=0m-1▵q0 twice at most. We obtain
(2.11)∑p=1T-1∑q=0m-1n(Ω¯pq,R̃z)≤(1+1M+1N)n(1-aT,R̃z).
Obviously, each Ω¯pq can be mapped conformally to the unit disc |ζ|<1 such that the center of Ω¯pq is mapped to ζ=0, and the image of Ωpq0 is contained in the disc |ζ|<η(<1). Since all Ω¯pq,Ωpq0 are similar, C is independent of p,q. Since S is conformally invariant, in view of Lemma 2.1, we obtain
(2.12)(2ν-1-2l)S(1-aT,w)≤(2ν-1-2l)∑p=P+1T-1∑q=0m-1S(Ωpq0,w)+(2ν-1-2l)S(1-a2,w)≤∑p=P+1T-1∑q=0m-1[∑j=12ν+1n¯l)(Ω¯pq,w=αj)+l+1ln(Ω¯pq,R̃z)+Cδ10(1-η)]+(2ν-1-2l)S(1-a2,w)≤3νTma-T(ρ+1-ε)+l+1l(1+1M+1N)n(1-aT,R̃z)+CT(a-Tρ/11)10+(2ν-1-2l)S(1-a2,w).
For sufficiently large integerT(=[log(1-r)/loga]),r∈[1-aT,1-aT+1). Thus we get
(2.13)(2ν-1-2l)S(1-1-ra,w)≤(2ν-1-2l)S(1-aT,w)≤1(1-r)ρ+1-(ε/2)+l+1l(1+1M+1N)n(r,R̃z)+C(11-r)10ρ/11+C,
where C is a constant.
For any integerT(=[log(1-r)/loga]), we have (1-aT)/(a-aT)∈(1/a,(1+a)/a). We can choose one a(>((l+1)/l)((2ν-2)/(2ν-1-(2/l)))∈(0,1) as l≥2ν+1) such that (l/(l+1))((2ν-1-(2/l))/(2ν-2))∈(1/a,(1+a)/a). For a certain sufficiently fixed large inter T, we have
(2.14)k=(1-11+a+a2+⋯+aT-1)-1=1-aTa-aT<ll+1⋅2ν-1-(2/l)2ν-2.
This yields the following:
(2.15)1-((1-t)/a)t=1a-(1a-1)1t≥1a-(1a-1)11-aT=1a(1-11+a+⋯+aT-1)=1ak,
where t∈[1-aT,1-aT+1).
Hence(2.16)∫1-aTrS(1-((1-t)/a),w)tdt=∫1-aTrS(1-((1-t)/a),w)1-((1-t)/a)1-((1-t)/a)tdt≥1ak∫1-aTrS(1-((1-t)/a),w)1-((1-t)/a)dt=1k∫1-aT-11-((1-r)/a)S(x,w)xdx.
Next, we deduce the following:
(2.17)∫1-aTr1(1-t)ρ+1-(ε/2)tdt≤11-aT∫1-aTr1(1-t)ρ+1-(ε/2)dt=-11-aT∫1-aTr1(1-t)ρ+1-(ε/2)d(1-t)≤11-aT⋅1ρ-(ε/2)⋅1(1-r)ρ-(ε/2),
when r≥t≥1-aT.
In view of1-t≤t for t∈[1-aT,1-aT+1), we have
(2.18)∫1-aTr1(1-t)10ρ/11tdt≤∫1-aTr1(1-t)(10ρ/11)+1dt≤110ρ/11⋅1(1-r)10ρ/11.
Dividing both sides of (2.13) by νt and integrating it from 1-aT to r, we have
(2.19)(2ν-1-2l)1ν∫1-aTrS(1-((1-t)/a),w)tdt≤1ν∫1-aTr1(1-t)ρ+1-(ε/2)tdt+l+1l(1+1M+1N)×[1ν∫1-aTrn(t,R̃z)-n(0,R̃z)tdt+n(0,R̃z)νlog⁡r-n(0,R̃z)νlog⁡(1-aT)]+Cν∫1-aTr1(1-t)10ρ/11tdt≤1ν⋅11-aT⋅1ρ-(ε/2)⋅1(1-r)ρ-(ε/2)+l+1l⋅(1+1M+1N)[1ν∫0rn(t,R̃z)-n(0,R̃z)tdt+n(0,R̃z)νlog⁡⁡r]-l+1l⋅(1+1M+1N)n(0,R̃z)νlog⁡(1-aT)+Cν110ρ/111(1-r)10ρ/11=1ν⋅1ρ-(ε/2)⋅1(1-r)ρ-(ε/2)+l+1l⋅(1+1M+1N)N(r,R̃z)-l+1l⋅(1+1M+1N)n(0,R̃z)νlog⁡(1-aT)+Cν110ρ/111(1-r)10ρ/11.
Note that T is fixed, we see that T(1-aT-1,w) is a finite constant.
Hence,(2.20)(2ν-1-2l)1k1ν∫1-aT-11-((1-r)/a)S(t,w)tdt≤(2ν-1-2l)1ν∫1-aTrS(1-((1-t)/a))tdt≤11-aT⋅1ν⋅1ρ-(ε/2)1(1-r)ρ-(ε/2)+l+1l(1+1M+1N)N(r,R̃z)-l+1l(1+1M+1N)n(0,R̃z)νlog⁡(1-aT)+Cν110ρ/111(1-r)10ρ/11.
Then
(2.21)(2ν-1-2l)1kν∫01-((1-r)/a)S(t,w)tdt≤11-aT⋅1ν1ρ-(ε/2)1(1-r)ρ-(ε/2)+l+1l(1+1M+1N)N(r,R̃z)-l+1l(1+1M+1N)n(0,R̃z)νlog⁡(1-aT)+Cν110ρ/111(1-r)10ρ/11+1kT(1-aT-1,w).
In view of [3], we know that
(2.22)N(r,R̃z)≤2(ν-1)T(r,w)+O(1).
We obtain
(2.23)(2ν-1-2l)⋅1kT(1-1-ra,w)≤11-aT⋅1ν(ρ-(ε/2))1(1-r)ρ-(ε/2)+l+1l⋅(1+1M+1N)2(ν-1)T(r,w)+Clog⁡⁡(1-aT)+Cν110ρ/111(1-r)10ρ/11+C,
where C is a constant.
Dividing both sides of the above inequality byU(1/(1-r))=(1/(1-r))ρ(1/(1-r)), we have
(2.24)(2ν-1-2l)⋅1kT(1-((1-r)/a),w)U(1/(1-r))≤11-aT⋅1ν(ρ-(ε/2))(1-r)ρ(1/(1-r))(1-r)ρ-(ε/2)+l+1l⋅(1+1M+1N)2(ν-1)T(r,w)U(1/(1-r))+Clog⁡⁡(1-aT)(1/(1-r))ρ(1/(1-r))+Cν110ρ/11(1-r)ρ(1/(1-r))(1-r)10ρ/11+C(1/(1-r))ρ(1/(1-r)).
We note that
(2.25)T(1-((1-r)/a),w)U(1/(1-r))=T(1-((1-r)/a),w)U(a/(1-r))U(a/(1-r))U(1/(1-r)).
In view of the properties of the U(1/(1-r)), we obtain
(2.26)limsup⁡r→1-T(1-((1-r)/a),w)U(1/(1-r))≥limsup⁡r→1-T(1-((1-r)/a),w)U(a/(1-r))liminf⁡r→1-U(a/(1-r))U(1/(1-r))=limsup⁡r→1-T(1-((1-r)/a),w)U(a/(1-r))lim⁡r→1-U(a/(1-r))U(1/(1-r))=aρ.
Letting r→1- in (2.24), we have
(2.27)(2ν-1-2l)1kaρ≤l+1l(1+1M+1N)2(ν-1),
that is,
(2.28)2ν-1-2l≤l+1l(1+1M+1N)2(ν-1)ka-ρ.
Letting a→1-,M→+∞, and N→+∞, we obtain
(2.29)k≥ll+12ν-1-(2/l)2ν-2.
This is contradictory to k<(l/(l+1))((2ν-1-(2/l))/(2ν-2)), and the lemma is proved.


## 3. Proof of the Theorem

Proof.
Chooseεn=ρ/2n,Rn=1-(1/2n).
In view of Lemma2.2, there exists an∈(1-(1/n),1),mn=[2π/(1-an)],pn,qn,θqn=(2π(qn)+1)/mn, and
(3.1)Ωpnqn={1-anpn≤|z|≤1-anpn+2}⋂{|arg⁡z-θqn|≤2πmn},(n=1,2,…).
Let
(3.2)θn=θqn,zn=(1-anpn)eiθn.
Then
(3.3)1>rn=|zn|=1-anpn>Rn=1-12n⟶1-(n⟶+∞),lim⁡n→+∞anpn=0.
Set
(3.4)Bn=[(1-anpn+2)-(1-anpn)]+(1-anpn+2)2πmn≤[(1-an2pn)-(1-anpn)]+(1-an2pn)2πmn=(1-anpn)anpn+(1-anpn)(1+anpn)2πmn≤(1-anpn)[anpn+(1+anpn)2πmn].
Take
(3.5)σn=anpn+(1+anpn)2πmn,
then
(3.6)σn⟶0(n⟶+∞).
Put
(3.7)Γn={|z-zn|<rnσn}.
Then
(3.8)Ωpnqn⊂Γn.
In view of Lemma 2.2, for each n,
(3.9)n¯l)(Γn∩▵,w=α)≥1anρ+1-εn,
except for those complex numbers contained in the union of 2ν spherical discs each with radius anpnρ/11=(1-rn)ρ/11. Theorem 1.1 is proved.Remark 3.1.
By using the same method, we can prove the existence of filling discs forK-quasimeromorphic mappings whose general case is carefully discussed in another paper.



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