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

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.

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.

Let 𝑤=𝑤(𝑧)(𝑧∈Δ) be the 𝜈-valued algebroid function defined by irreducible equation𝐴_𝜈(𝑧)𝑤^𝜈+𝐴_𝜈−1(𝑧)𝑤^𝜈−1+⋯+𝐴₀(𝑧)=0, (1.1) where 𝐴_𝜈(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 𝑅_𝑧. A point in 𝑅_𝑧, whose projection in the 𝑧-plane is 𝑧, is denoted by 𝑧. The part of 𝑅_𝑧, which covers the disc {𝑧∶|𝑧|<𝑟}, is denoted by |𝑧|<𝑟.

Denote𝑆()=1𝑟,𝑤𝜋_|𝑧_|≤𝑟|𝑤^′(𝑧)|1+|𝑤(𝑧)|²²𝑑𝜔. (1.2)𝑆(𝑟,𝑤)is called the mean covering number of |𝑧|≤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 |𝑧|≤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.

Let𝑁()=1𝑟,𝑎𝜈^𝑟₀𝑛(𝑡,𝑎)−𝑛(0,𝑎)𝑡𝑛()𝑑𝑡+0,𝑎𝜈𝑚()=1log𝑟,𝑟,𝑎2𝜋𝜈_|𝑧^𝜈_|=𝑟𝑗=1log⁺1𝑤_𝑗𝑟𝑒^𝑖𝜃−𝑎𝑑𝜃,𝑧=𝑟𝑒^𝑖𝜃, (1.3) where |𝑧|=𝑟 is the boundary of |𝑧|≤𝑟. The characteristic function of 𝑤(𝑧) is defined by𝑇()=1𝑟,𝑤𝜈^𝑟₀𝑆(𝑡,𝑤)𝑡𝑑𝑡. (1.4) In view of [4], we have𝑇(𝑟,𝑤)=𝑚(𝑟,𝑤)+𝑁(𝑟,∞)+𝑂(1). (1.5) The order of algebroid function 𝑤(𝑧) is defined by𝜌=limsup_𝑟→1−log𝑇(𝑟,𝑤)((.log1/1−𝑟)) (1.6)

In this paper we assume that 0<𝜌<+∞,𝑉 is the 𝑤-sphere, and 𝐶 is a constant which can stand for different constant. Let 𝑛(𝑟,𝑅_𝑧) be the number of the branch points of 𝑅_𝑧 in |𝑧|≤𝑟, counted with the order of branch. Write𝑁𝑟,𝑅_𝑧=1𝜈^𝑟₀𝑛𝑡,𝑅_𝑧−𝑛0,𝑅_𝑧𝑡𝑛𝑅𝑑𝑡+0,_𝑧𝜈log𝑟. (1.7)

Valiron is the first one to introduce the concept of a proximate order 𝜌(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, andlim_𝑟→1−𝜌11−𝑟=𝜌,lim_𝑟→1−𝑈(𝑘/(1−𝑟))𝑈(1/(1−𝑟))=𝑘^𝜌𝑘isanygivenpositiveconstant,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. (1.8)

For an algebroid function 𝑤(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 that 𝑤(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 Γ_𝑛∶𝑧−𝑧_𝑛<𝑟_𝑛𝜎_𝑛,𝑛=1,2,…, (1.9) where 𝑧_𝑛=𝑟_𝑛𝑒^𝑖𝜃𝑛,lim_𝑛→∞𝑟_𝑛=1,𝜎_𝑛>0,lim_𝑛→∞𝜎_𝑛=0. (1.10) Such that for each 𝛼𝑛^𝑙)Γ_𝑛≥1∩▵,𝑤=𝛼1−𝑟_𝑛^{𝜌+1−𝜀𝑛}, (1.11) except for those complex numbers contained in the union of 2𝜈 spherical discs each with radius (1−𝑟_𝑛)^𝜌/11, where lim_𝑛→+∞𝜀_𝑛=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(Γ_𝑛∩▵,𝑤=𝛼)≥1/(1−𝑟_𝑛)^{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 that 𝑤(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,…,𝑎_𝑞(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 2𝑞−2−𝑙𝑆(𝑟,𝑤)≤^𝑞_𝑗=1𝑛^𝑙)𝑅,𝑎_𝑗+𝑙+1𝑙𝑛𝑅,𝑅_𝑧+𝐶𝑅(𝑅−𝑟)𝛿¹⁰. (2.1)

Combining the potential theory with Lemma 2.1, one proves Lemma 2.2, which is crucial to the theorem.

Lemma 2.2. Suppose that 𝑤(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 𝑟_𝑛=1−𝑎^𝑛,𝑚=2𝜋1−𝑎,𝜃_𝑞=2𝜋𝑞+1𝑚,Ω_𝑝𝑞=1−𝑎^𝑝≤|𝑧|<1−𝑎^𝑝+2∩arg𝑧−𝜃_𝑞≤2𝜋𝑚𝑝=1,2,3,…;𝑞=0,1,2,…,𝑚−1, (2.2) where [𝑥] stands for the inter part of 𝑥.
Then, among 𝑝,𝑞, there exists at least one pair 𝑝₀,𝑞₀, such that 1−𝑎^𝑝0>𝑅, and in Ω_𝑝0𝑞0, 𝑛^𝑙)Ω_𝑝0𝑞0≥1,𝑤=𝛼𝑎^{𝜌+1−𝜀}, (2.3) except for those complex numbers contained in the union of 2𝜈 spherical discs each with radius 𝛿=𝑎^𝑝0𝜌/11.

Proof. Suppose the conclusion is false. Then there exists a sequence {𝑎_𝑖}^∞_𝑖=1(0<𝑎_𝑖<1), where lim_𝑖→∞𝑎_𝑖=1. For any 𝑎∈{𝑎_𝑖}, 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=𝑎^𝑝𝜌/40. Denote 𝛼_𝑗=𝛼_𝑗𝑝,𝑞^2𝜈+1_𝑗=1. (2.4) For any 𝑝,𝑞 mentioned above, we have 𝑛^𝑙)Ω_𝑝𝑞,𝑤=𝛼_𝑗<1𝑎^{𝑝(𝜌+1−𝜀)}. (2.5)
For any 𝑟>𝑅, let 𝑇=[log(1−𝑟)/log𝑎], then we have 1−𝑎^𝑇≤𝑟<1−𝑎^𝑇+1.
For any given positive integers 𝑁 and 𝑀, set 𝑏=𝑎^1/𝑀∈(0,1),𝛾_𝑝𝑡=1−𝑏^{𝑀𝑝+𝑡}𝐿,𝑡=0,1,2,…,𝑀−1,_𝑝𝑡=𝛾_𝑝𝑡≤|𝑧|<𝛾_𝑝,𝑡+1,𝜃_𝑞𝑗=2𝜋𝑞𝑚+2𝜋𝑗,▵𝑁𝑚_𝑞𝑗=𝑧∶|𝑧|<1−𝑎^𝑇,𝜃_𝑞𝑗≤arg𝑧<𝜃_𝑞,𝑗+1. (2.6) Then 1−𝑎≤|𝑧|<1−𝑎^𝑇=^𝑀−1_𝑡=0^𝑇−1_𝑝=1𝐿_𝑝𝑡,|𝑧|<1−𝑎^𝑇=^𝑁−1_𝑗=0^𝑚−1_𝑞=0▵_𝑞𝑗. (2.7) Thus there exists 𝑡₀,𝑗₀ which are related to 𝑇. We can assume 𝑡₀=0,𝑗₀=0, such that ^𝑇−1_𝑝=1𝑛𝐿_𝑝0,𝑅_𝑧≤1𝑀𝑛1−𝑎^𝑇,𝑅_𝑧,_{𝑚−1𝑞=0}𝑛▵_𝑞0,𝑅_𝑧≤1𝑁𝑛1−𝑎^𝑇,𝑅_𝑧. (2.8) Set Ω⁰_𝑝𝑞=𝑏1−^𝑀𝑝+𝑏^𝑀𝑝+12≤|𝑧|𝑏<1−^{𝑀𝑝+𝑀}+𝑏^{𝑀𝑝+𝑀+1}2∩𝜃_𝑞0+𝜃_𝑞12𝜃≤arg𝑧<^𝑞+1,0+𝜃_𝑞+1,12,Ω_𝑝𝑞=1−𝑏^𝑀𝑝≤|𝑧|<1−𝑏^{𝑀𝑝+𝑀+1}∩𝜃_𝑞0≤arg𝑧<𝜃_𝑞+1,1. (2.9) Then we have Ω⁰_𝑝𝑞⊂Ω_𝑝𝑞⊂Ω_𝑝𝑞. (2.10) Since {Ω_𝑝𝑞}_𝑝,𝑞 covers ⋃^𝑇−1_𝑝=1𝐿_𝑝0 and ⋃^𝑚−1_𝑞=0▵_𝑞0 twice at most. We obtain ^𝑇−1_𝑝=1^𝑚−1_𝑞=0𝑛Ω_𝑝𝑞,𝑅_𝑧≤11+𝑀+1𝑁𝑛1−𝑎^𝑇,𝑅_𝑧. (2.11) 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 Ω⁰_𝑝𝑞 is contained in the disc |𝜁|<𝜂(<1). Since all Ω_𝑝𝑞,Ω⁰_𝑝𝑞 are similar, 𝐶 is independent of 𝑝,𝑞. Since 𝑆 is conformally invariant, in view of Lemma 2.1, we obtain 22𝜈−1−𝑙𝑆1−𝑎^𝑇≤2,𝑤2𝜈−1−𝑙^𝑇−1_𝑝=𝑃+1^𝑚−1_𝑞=0𝑆Ω⁰_𝑝𝑞+2,𝑤2𝜈−1−𝑙𝑆1−𝑎²≤,𝑤_{𝑇−1𝑝=𝑃+1𝑚−1𝑞=02𝜈+1𝑗=1}𝑛^𝑙)Ω_𝑝𝑞,𝑤=𝛼_𝑗+𝑙+1𝑙𝑛Ω_𝑝𝑞,𝑅_𝑧+𝐶𝛿¹⁰+21−𝜂2𝜈−1−𝑙𝑆1−𝑎²,𝑤≤3𝜈𝑇𝑚𝑎^{−𝑇(𝜌+1−𝜀)}+𝑙+1𝑙11+𝑀+1𝑁𝑛1−𝑎^𝑇,𝑅_𝑧+𝐶𝑇𝑎^{−𝑇𝜌/1110}+22𝜈−1−𝑙𝑆1−𝑎²,𝑤. (2.12)
For sufficiently large integer 𝑇(=[log(1−𝑟)/log𝑎]),𝑟∈[1−𝑎^𝑇,1−𝑎^𝑇+1). Thus we get 22𝜈−1−𝑙𝑆1−1−𝑟𝑎≤2,𝑤2𝜈−1−𝑙𝑆1−𝑎^𝑇≤1,𝑤(1−𝑟)^{𝜌+1−(𝜀/2)}+𝑙+1𝑙11+𝑀+1𝑁𝑛𝑟,𝑅_𝑧1+𝐶1−𝑟^10𝜌/11+𝐶, (2.13) where 𝐶 is a constant.
For any integer 𝑇(=[log(1−𝑟)/log𝑎]), we have (1−𝑎^𝑇)/(𝑎−𝑎^𝑇)∈(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 1𝑘=1−1+𝑎+𝑎²+⋯+𝑎^𝑇−1−1=1−𝑎^𝑇𝑎−𝑎^𝑇<𝑙⋅()𝑙+12𝜈−1−2/𝑙.2𝜈−2 (2.14) This yields the following: 1−((1−𝑡)/𝑎)𝑡=1𝑎−1𝑎1−1𝑡≥1𝑎−1𝑎1−11−𝑎^𝑇=1𝑎11−1+𝑎+⋯+𝑎^𝑇−1=1,𝑎𝑘 (2.15) where 𝑡∈[1−𝑎^𝑇,1−𝑎^𝑇+1).
Hence ^𝑟_1−𝑎𝑇𝑆(1−((1−𝑡)/𝑎),𝑤)𝑡𝑑𝑡=^𝑟_1−𝑎𝑇𝑆(1−((1−𝑡)/𝑎),𝑤)))))1−((1−𝑡/𝑎1−((1−𝑡/𝑎𝑡≥1𝑑𝑡𝑎𝑘^𝑟_1−𝑎^𝑇𝑆(1−((1−𝑡)/𝑎),𝑤)))=11−((1−𝑡/𝑎𝑑𝑡𝑘^{1−((1−𝑟)/𝑎)}_1−𝑎^𝑇−1𝑆(𝑥,𝑤)𝑥𝑑𝑥. (2.16) Next, we deduce the following: ^𝑟_1−𝑎𝑇1(1−𝑡)^{𝜌+1−(𝜀/2)}𝑡1𝑑𝑡≤1−𝑎^𝑇𝑟_1−𝑎^𝑇1(1−𝑡)^{𝜌+1−(𝜀/2)}1𝑑𝑡=−1−𝑎^𝑇𝑟_1−𝑎^𝑇1(1−𝑡)^{𝜌+1−(𝜀/2)}𝑑()≤11−𝑡1−𝑎^𝑇⋅1()⋅1𝜌−𝜀/2(1−𝑟)^{𝜌−(𝜀/2)}, (2.17) when 𝑟≥𝑡≥1−𝑎^𝑇.
In view of 1−𝑡≤𝑡 for 𝑡∈[1−𝑎^𝑇,1−𝑎^𝑇+1), we have ^𝑟_1−𝑎𝑇1(1−𝑡)^10𝜌/11𝑡𝑑𝑡≤^𝑟_1−𝑎𝑇1(1−𝑡)^{(10𝜌/11)+1}1𝑑𝑡≤⋅110𝜌/11(1−𝑟)^10𝜌/11. (2.18) Dividing both sides of (2.13) by 𝜈𝑡 and integrating it from 1−𝑎^𝑇 to 𝑟, we have 22𝜈−1−𝑙1𝜈^𝑟_1−𝑎^𝑇𝑆(1−((1−𝑡)/𝑎),𝑤)𝑡≤1𝑑𝑡𝜈^𝑟_1−𝑎^𝑇1(1−𝑡)^{𝜌+1−(𝜀/2)}𝑡𝑑𝑡+𝑙+1𝑙11+𝑀+1𝑁×1𝜈^𝑟_1−𝑎^𝑇𝑛𝑡,𝑅_𝑧−𝑛0,𝑅_𝑧𝑡𝑛𝑅𝑑𝑡+0,_𝑧𝜈𝑛𝑅log𝑟−0,_𝑧𝜈log1−𝑎^𝑇+𝐶𝜈^𝑟_1−𝑎^𝑇1(1−𝑡)^10𝜌/11𝑡1𝑑𝑡≤𝜈⋅11−𝑎^𝑇⋅1()⋅1𝜌−𝜀/2(1−𝑟)^{𝜌−(𝜀/2)}+𝑙+1𝑙⋅11+𝑀+1𝑁1𝜈^𝑟₀𝑛𝑡,𝑅_𝑧−𝑛0,𝑅_𝑧𝑡𝑛𝑅𝑑𝑡+0,_𝑧𝜈−log𝑟𝑙+1𝑙⋅11+𝑀+1𝑁𝑛0,𝑅_𝑧𝜈log1−𝑎^𝑇+𝐶𝜈1110𝜌/11(1−𝑟)^10𝜌/11=1𝜈⋅1()⋅1𝜌−𝜀/2(1−𝑟)^{𝜌−(𝜀/2)}+𝑙+1𝑙⋅11+𝑀+1𝑁𝑁𝑟,𝑅_𝑧−𝑙+1𝑙⋅11+𝑀+1𝑁𝑛0,𝑅_𝑧𝜈log1−𝑎^𝑇+𝐶𝜈1110𝜌/11(1−𝑟)^10𝜌/11. (2.19) Note that 𝑇 is fixed, we see that 𝑇(1−𝑎^𝑇−1,𝑤) is a finite constant.
Hence, 22𝜈−1−𝑙1𝑘1𝜈^{1−((1−𝑟)/𝑎)}_1−𝑎^𝑇−1𝑆(𝑡,𝑤)𝑡≤2𝑑𝑡2𝜈−1−𝑙1𝜈^𝑟_1−𝑎^𝑇𝑆(1−((1−𝑡)/𝑎))𝑡≤1𝑑𝑡1−𝑎^𝑇⋅1𝜈⋅1()1𝜌−𝜀/2(1−𝑟)^{𝜌−(𝜀/2)}+𝑙+1𝑙11+𝑀+1𝑁𝑁𝑟,𝑅_𝑧−𝑙+1𝑙11+𝑀+1𝑁𝑛0,𝑅_𝑧𝜈log1−𝑎^𝑇+𝐶𝜈1110𝜌/11(1−𝑟)^10𝜌/11. (2.20) Then 22𝜈−1−𝑙1𝑘𝜈₀^{1−((1−𝑟)/𝑎)}𝑆(𝑡,𝑤)𝑡1𝑑𝑡≤1−𝑎^𝑇⋅1𝜈1()1𝜌−𝜀/2(1−𝑟)^{𝜌−(𝜀/2)}+𝑙+1𝑙11+𝑀+1𝑁𝑁𝑟,𝑅_𝑧−𝑙+1𝑙11+𝑀+1𝑁𝑛0,𝑅_𝑧𝜈log1−𝑎^𝑇+𝐶𝜈1110𝜌/11(1−𝑟)^10𝜌/11+1𝑘𝑇1−𝑎^𝑇−1,𝑤. (2.21) In view of [3], we know that 𝑁𝑟,𝑅_𝑧≤2(𝜈−1)𝑇(𝑟,𝑤)+𝑂(1). (2.22) We obtain 22𝜈−1−𝑙⋅1𝑘𝑇1−1−𝑟𝑎≤1,𝑤1−𝑎^𝑇⋅1𝜈()1𝜌−𝜀/2(1−𝑟)^{𝜌−(𝜀/2)}+𝑙+1𝑙⋅11+𝑀+1𝑁2(𝜈−1)𝑇(𝑟,𝑤)+𝐶log1−𝑎^𝑇+𝐶𝜈1110𝜌/11(1−𝑟)^10𝜌/11+𝐶, (2.23) where 𝐶 is a constant.
Dividing both sides of the above inequality by 𝑈(1/(1−𝑟))=(1/(1−𝑟))^{𝜌(1/(1−𝑟))}, we have 22𝜈−1−𝑙⋅1𝑘𝑇(1−((1−𝑟)/𝑎),𝑤)𝑈((≤11/1−𝑟))1−𝑎^𝑇⋅1𝜈()()𝜌−𝜀/21−𝑟^{𝜌(1/(1−𝑟))}(1−𝑟)^{𝜌−(𝜀/2)}+𝑙+1𝑙⋅11+𝑀+1𝑁2()𝑇()𝜈−1𝑟,𝑤𝑈(1/(1−𝑟))+𝐶log1−𝑎^𝑇(1/(1−𝑟))^{𝜌(1/(1−𝑟))}+𝐶𝜈1()10𝜌/111−𝑟^{𝜌(1/(1−𝑟))}(1−𝑟)^10𝜌/11+𝐶(1/(1−𝑟))^{𝜌(1/(1−𝑟))}. (2.24) We note that 𝑇(1−((1−𝑟)/𝑎),𝑤)𝑈((=𝑇()))1/1−𝑟))1−((1−𝑟/𝑎,𝑤𝑈((𝑈((𝑎/1−𝑟))𝑎/1−𝑟))𝑈((.1/1−𝑟)) (2.25) In view of the properties of the 𝑈(1/(1−𝑟)), we obtain 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))=𝑎^𝜌. (2.26) Letting 𝑟→1− in (2.24), we have 22𝜈−1−𝑙1𝑘𝑎^𝜌≤𝑙+1𝑙11+𝑀+1𝑁2(𝜈−1), (2.27) that is, 22𝜈−1−𝑙≤𝑙+1𝑙11+𝑀+1𝑁2(𝜈−1)𝑘𝑎^−𝜌. (2.28) Letting 𝑎→1−,𝑀→+∞, and 𝑁→+∞, we obtain 𝑙𝑘≥()𝑙+12𝜈−1−2/𝑙.2𝜈−2 (2.29) 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. Choose 𝜀_𝑛=𝜌/2𝑛,𝑅_𝑛=1−(1/2^𝑛).
In view of Lemma 2.2, there exists 𝑎_𝑛∈(1−(1/𝑛),1),𝑚_𝑛=[2𝜋/(1−𝑎_𝑛)],𝑝_𝑛,𝑞_𝑛,𝜃_𝑞𝑛=(2𝜋(𝑞_𝑛)+1)/𝑚_𝑛, and Ω_{𝑝𝑛𝑞𝑛}=1−𝑎^𝑝𝑛_𝑛≤|𝑧|≤1−𝑎^𝑝𝑛_𝑛⁺²arg𝑧−𝜃_𝑞𝑛≤2𝜋𝑚_𝑛,(𝑛=1,2,…). (3.1) Let 𝜃_𝑛=𝜃_𝑞𝑛,𝑧_𝑛=1−𝑎^𝑝𝑛_𝑛𝑒^𝑖𝜃𝑛. (3.2) Then 1>𝑟_𝑛=𝑧_𝑛=1−𝑎^𝑝𝑛_𝑛>𝑅_𝑛1=1−2^𝑛⟶1−(𝑛⟶+∞),lim_𝑛→+∞𝑎^𝑝𝑛_𝑛=0. (3.3) Set 𝐵_𝑛=1−𝑎^𝑝𝑛_𝑛⁺²−1−𝑎^𝑝𝑛_𝑛+1−𝑎^𝑝𝑛_𝑛⁺²2𝜋𝑚_𝑛≤1−𝑎^2𝑝𝑛_𝑛−1−𝑎^𝑝𝑛_𝑛+1−𝑎^2𝑝𝑛_𝑛2𝜋𝑚_𝑛=1−𝑎^𝑝𝑛_𝑛𝑎^𝑝𝑛_𝑛+1−𝑎^𝑝𝑛_𝑛1+𝑎^𝑝𝑛_𝑛2𝜋𝑚_𝑛≤1−𝑎^𝑝𝑛_𝑛𝑎^𝑝𝑛_𝑛+1+𝑎^𝑝𝑛_𝑛2𝜋𝑚_𝑛. (3.4) Take 𝜎_𝑛=𝑎^𝑝𝑛_𝑛+1+𝑎^𝑝𝑛_𝑛2𝜋𝑚_𝑛, (3.5) then 𝜎_𝑛⟶0(𝑛⟶+∞). (3.6) Put Γ_𝑛=𝑧−𝑧_𝑛<𝑟_𝑛𝜎_𝑛. (3.7) Then Ω_{𝑝𝑛𝑞𝑛}⊂Γ_𝑛. (3.8) In view of Lemma 2.2, for each 𝑛, 𝑛^𝑙)Γ_𝑛≥1∩▵,𝑤=𝛼𝑎^{𝜌+1−𝜀𝑛}_𝑛, (3.9) except for those complex numbers contained in the union of 2𝜈 spherical discs each with radius 𝑎^𝑝𝑛_𝑛^𝜌/11=(1−𝑟_𝑛)^𝜌/11. Theorem 1.1 is proved.

Remark 3.1. By using the same method, we can prove the existence of filling discs for 𝐾-quasimeromorphic mappings whose general case is carefully discussed in another paper.

Acknowledgments

The authors would like to thank the referee for his/her many helpful suggestions on an early version of the paper. Z.-X. Zuxing is supported in part by Beijing Municipal Research Foundation for The Excellent Scholars Ministry (2011D005022000009); Science and Technology Research Program of Beijng Municipal Commission of Education (KM201211417011); Funding Project for Academic Human Resources Development in Beijing Union University.