A Fundamental Inequality of Algebroidal Function

Abstract

By using a new mapping of Ahlfors covering surfaces, a fundamental inequality in the angular domain for the algebroidal function is obtained.

1. Introduction and Main Results

In the field of valued distribution, the fundamental inequality is an important tool. For example, it can be used to investigate the singular direction [1]. Using geometric theory, Tsuji firstly obtained the second fundamental theory in an angular domain and proved the existence of Borel direction [2]. The value distribution theory of meromorphic functions was extended to algebroidal functions last century [3]. In 1983, Lv and Gu proved an inequality of algebroidal function for an angular domain [4]. By the inequality, some results of singular direction are obtained; see [5, 6]. In [7], the authors obtained a more accurate inequality for angular domain. In this paper, we will use a new method to simplify and extent an inequality of Tsuji to algebroidal functions.

First, we recall some definitions from [3].

Suppose that 𝐴_𝑣(𝑧),…,𝐴₀(𝑧) are analytic functions with no common zeros in the complex plane. Ψ(𝑧,𝑊) is a bivariate complex function and satisfies Ψ(𝑧,𝑊)=𝐴_𝑣(𝑧)𝑊^𝑣+𝐴_𝑣−1(𝑧)𝑊^𝑣−1+⋯+𝐴₀(𝑧)=0. (1.1) For all 𝑧 in the complex plane, the equation Ψ(𝑧,𝑊)=0 has 𝑣 complex roots 𝑤₁(𝑧),𝑤₂(𝑧),…,𝑤_𝑣(𝑧). Then, (1.1) defines a 𝑣-valued algebroidal function 𝑊(𝑧); see [3, 8]. If 𝐴_𝑣(𝑧)=1, then 𝑊(𝑧) is called 𝑣-valued integral algebroidal function. If Ψ(𝑧,𝑊) is irreducible, correspondingly 𝑊(𝑧) is called 𝑣-valued irreducible algebroidal function (note that 𝑊(𝑧) is a meromorphic function, if 𝑣=1). Now we suppose that 𝑊(𝑧) is an irreducible algebriodal function defined by (1.1).

If 𝐴_𝑣(𝑧₀)≠0, and the 𝑘-degree equation Ψ(𝑧₀,𝑊)=0 and its partial derivative (𝜕Ψ/𝜕𝑊)(𝑧₀,𝑊)=0 have no common roots (i.e., 𝑧₀ is not a multiple root of Ψ(𝑧₀,𝑊)=0), then 𝑧₀ is said to be a regular point. The set of all regular points is called the regular set, denoted by 𝑇_𝑊. Its complementary set 𝑆_𝑊 : ={𝑧∣|𝑧|<∞}−𝑇_𝑊 is called the critical set. Obviously, 𝑆_𝑤 includes all branch points of 𝑊(see [3]).

The domain of a 𝑣-valued irreducible algebriodal function 𝑊 is a connected Riemann surface [8], and its single-valued domain is denoted by 𝑅_𝑧. A point in 𝑅_𝑧 is 𝑧 and sets lying over |𝑧|<𝑟 and {𝜙₁<arg𝑧<𝜙₂}(𝜙₁<𝜙₂) are |𝑧|<𝑟 and Ω(𝜙₁,𝜙₂). Let 𝑛(𝑟,𝑊=𝑎) and 𝑛(Ω(𝜙₁,𝜙₂),𝑟,𝑊=𝑎) be the number of zeros, counted according to their multiplicities, of 𝑊=𝑎 in {|𝑧|<𝑟} and {|𝑧|<𝑟}⋂Ω(𝜙₁,𝜙₂), respectively. Let 𝑛(𝑟,𝑊=𝑎) be the number of distinct zeros in {|𝑧|<𝑟}, and let 𝑛(𝑟,𝑅_𝑧) be the number of branch points in {|𝑧|<𝑟}. Similarly, we can define 𝑛(Ω(𝜙₁,𝜙₂),𝑟,𝑊=𝑎) and 𝑛(Ω(𝜙₁,𝜙₂),𝑟,𝑅_𝑧). Let 𝑆()=1𝑟,𝑊𝜋_|𝑧_|≤𝑟𝑊^′(𝑧)1+|𝑊(𝑧)|²²𝑑𝑊𝑧=𝑟𝑒^𝑖𝜃,𝑆Ω𝜙₁,𝜙₂=1,𝑟,𝑊𝜋_{|𝑧_⋂|≤𝑟}Ω_{(𝜙1,𝜙2)}|𝑊^′(𝑧)|1+|𝑊(𝑧)|²²𝑇()=1𝑑𝑊,𝑟,𝑊𝑣^𝑟₀𝑆(𝑡,𝑊)𝑡𝑇Ω𝜙𝑑𝑡,₁,𝜙₂=1,𝑟,𝑊𝑣^𝑟₀𝑆Ω𝜙₁,𝜙₂,𝑡,𝑊𝑡𝑁()=1𝑑𝑡,𝑟,𝑊=𝑎𝑣^𝑟₀𝑛(𝑡,𝑊=𝑎)−𝑛(0,𝑊=𝑎)𝑡𝑛()𝑑𝑡+0,𝑊=𝑎𝑣𝑁𝑅ln𝑟,𝑟,_𝑧=1𝑣^𝑟₀𝑛𝑡,𝑅_𝑧−𝑛0,𝑅_𝑧𝑡𝑛𝑅𝑑𝑡+0,_𝑧𝑣𝑚()=1ln𝑟,𝑟,𝑊2𝜋𝑣^𝑣_𝑘=10^2𝜋ln⁺𝑤_𝑘𝑟𝑒^𝑖𝜃𝑑𝜃. (1.2)

Similarly, we can define 𝑁(Ω(𝜙₁,𝜙₂),𝑟,𝑊=𝑎), 𝑁(Ω(𝜙₁,𝜙₂),𝑟,𝑊=𝑎) and 𝑁(Ω(𝜙₁,𝜙₂),𝑟,𝑅_𝑧). From [3], we know that 𝑇(𝑟,𝑤)=𝑁(𝑟,𝑊=∞)+𝑚(𝑟,𝑊)+𝑂(1) and 𝑁(𝑟,𝑅_𝑧)≤2(𝑣−1)𝑇(𝑟,𝑊)+𝑂(1).

In this paper, we will prove the main theorem.

Theorem 1.1. Let 𝑊(𝑧) be a 𝑣-valued algebroidal function in region Ω(𝜙₁,𝜙₂)≜{|𝑧|∣𝜙₁<arg𝑧<𝜙₂}(𝜙₁<𝜙₂). 𝑎₁,𝑎₂,…,𝑎_𝑞(𝑞⩾3) are 𝑞 different complex numbers on the sphere with radius 𝛿∈(0,1/2). For 𝜙,𝜀^∗,𝜀(0<𝜀^∗<𝜀,𝜙₁<𝜙−𝜀<𝜙−𝜀^∗<𝜙+𝜀^∗<𝜙+𝜀<𝜙₂), and 𝑅>𝑅^∗>2, we have 𝑞−2𝑆Ω𝜙−𝜀^∗,𝜙+𝜀^∗,𝑅^∗Ω𝑅,𝑊⩽𝑛𝜙−𝜀,𝜙+𝜀,𝑅,_𝑧+^𝑞_𝑗=1𝑛Ω𝜙−𝜀,𝜙+𝜀,𝑅,𝑊=𝑎_𝑗+2⁵⁶𝑣𝜋²⁴ln𝑅𝛿³⁸(𝜀−𝜀^∗)(ln𝑅−ln𝑅^∗)+𝑞−2𝑆Ω𝜙−𝜀^∗,𝜙+𝜀^∗,1𝑅^∗,𝑊. (1.3)

By the inequality in Theorem 1.2, we will immediately have the following.

Theorem 1.2. For a meromorphic function 𝑊 (a 1-valued algebroidal function with no branch points) defined by (1.1) satisfying lim_𝑅→∞𝑇(𝑟,𝑊)ln²𝑅=∞, (1.4) it has at least one Nevanlinna direction, that is, there exists arg𝑧=𝜙₀, such that ∑_{⋃𝑎∈𝐶{∞}}𝛿(𝑎,𝜙₀)≤2 holds for any finitely many deficient value 𝑎, where 𝛿𝑎,𝜙₀=1−lim_𝜀→0+lim_𝑅→∞𝑁Ω𝜙₀−𝜀,𝜙₀+𝜀,𝑅,𝑊=𝑎𝑇Ω𝜙₀−𝜀,𝜙₀+𝜀,𝑅,𝑊>0. (1.5)

2. Some Lemmas

First, it is easy to prove the following.

Lemma 2.1. Suppose that 𝑎,𝑏>0, then there exists 𝑝,𝑞>0, such that 𝑝+𝑖𝑞=1,𝑎−𝑏𝑖 (2.1) where 𝑝=𝑎²+𝑏²+𝑎2𝑎²+𝑏²,𝑞=𝑎²+𝑏²−𝑎2𝑎²+𝑏². (2.2)

Lemma 2.2. Suppose that 𝐴(𝑧)=∫^𝑧₀(𝑑𝑡/𝑡(1−𝑡²)) and 𝐵(𝑧)=((1−𝑧)/(1+𝑧))𝑖, then (1)the mapping 𝐺₁=𝐴∘𝐵(𝑧) maps the unit disc 𝑈 to the square 𝑄≜{𝑧=𝑥+𝑖𝑦∣0<𝑥<2ℎ,0<𝑦<2ℎ}, where ℎ is constant; (2)𝐺₁ maps {|𝑧|<𝑟}, where 0<𝑟<1, into a symmetrical convex region in 𝑄; (3)ℎ in Lemma 2.2 satisfies ℎ=¹_√2−1𝑑𝑡𝑡1−𝑡²=^√₀²⁻¹𝑑𝑡𝑡1−𝑡²>1. (2.3)

Proof. Obviously, (1) holds. By the definition of 𝐵(𝑧), we have 𝐵(1)=0,𝐵(−1)=∞,𝐵(𝑖)=1,𝐵(−𝑖)=−1.
In order to compute ℎ, first we prove that 𝐺₁ maps 𝑑(0), 𝑑(𝜋/2), 𝑑(𝜋/4), 𝑑(−𝜋/4) to four symmetry axes of Q, where 𝑑(𝜃)={𝑟𝑒^𝑖𝜃;−1<𝑟<1}. Let 𝑧=𝑟𝑒^𝑖𝜃 (𝜃 is fixed), 𝑟∈(−1,1), 𝐺₁(𝑧)=𝐴∘𝐵(𝑧)=^𝐵₀d𝑡𝑡1−𝑡²=−(1+𝑖)^𝑟₁𝑒^𝑖𝜃d𝑟1−𝑟⁴𝑒^𝑖4𝜃. (2.4)
Hence, when 𝜃=0, 𝜋/4, 𝜋/2, −𝜋/4, arg𝐺₁(𝑟𝑒^𝑖𝜃)=𝜋/4,𝜋/2,−𝜋/4,0, respectively, see Figure 1.
Then, we compute ℎ. Since 𝑧=0 is the only intersection point of the lines 𝑑(0) and 𝑑(𝜋/4). The center of the square 𝑄, ℎ+ℎ𝑖, is that of the curves 𝐺₁(𝑑(0)) and 𝐺₁(𝑑(𝜋/4)). Then, 𝐺₁ conforms 0 onto ℎ+ℎ𝑖.
Hence, ℎ=𝐺₁𝑒^𝑖𝜋/4=𝐴√2−1=^√₀²⁻¹𝑑𝑡𝑡1−𝑡²=¹_√2−1𝑑𝑡𝑡1−𝑡²>¹_√2−1𝑑𝑡√𝑡>1. (2.5)
(2) At last we prove that 𝐺₁({|𝑧|<𝑟}) is a convex region, see Figure 1.
For a fixed 𝑟∈(0,1), by (2.4), 𝐺₁𝑟𝑒^𝑖𝜃=^𝑧₁−(1+𝑖)d𝑧√1−𝑧⁴=(1+𝑖)¹_𝑟𝑑𝑟√1−𝑟⁴+^𝜃₀(1−𝑖)𝑑𝜃𝑟⁻²−𝑟²cos2𝜃−𝑖𝑟⁻²+𝑟².sin2𝜃 (2.6)
Set 𝐺₁(𝑟𝑒^𝑖𝜃)=𝑥(𝜃)+𝑖𝑦(𝜃). When 𝜃∈(0,𝜋/4), cos2𝜃, sin2𝜃>0, by Lemma 2.1, 𝜕𝐺₁=𝜕𝜃1−𝑖𝑟⁻²−𝑟²cos2𝜃−𝑖𝑟⁻²−𝑟²=()=−sin2𝜃1−𝑖𝑝+𝑖𝑞𝑝+𝑞𝑝−𝑞𝑖≜𝑥^′+𝑦^′𝑖, (2.7) where 𝑝>𝑞>0. 𝜕²𝐺₁𝜕𝜃²=(1−𝑖)𝑟⁻²−𝑟²sin2𝜃+𝑖𝑟⁻²+𝑟²cos2𝜃𝑟⁻²−𝑟²cos2𝜃−𝑖𝑟⁻²+𝑟²sin2𝜃³=(1−𝑖)𝑝+𝑞𝑖𝑟⁻²−𝑟²sin2𝜃+𝑖𝑟⁻²+𝑟²cos2𝜃𝑟⁻²−𝑟²cos2𝜃−𝑖𝑟⁻²+𝑟²=𝑟sin2𝜃𝑝−𝑞⁻⁴−𝑟⁴−2𝑝+𝑞sin4𝜃𝑟⁻²−𝑟²²cos²2𝜃+𝑟⁻²+𝑟²²sin²2𝑟2𝜃+𝑖𝑝−𝑞sin4𝜃+𝑝+𝑞⁻⁴−𝑟⁴𝑟⁻²−𝑟²²cos²2𝜃+𝑟⁻²+𝑟²²sin²2𝜃=𝑥^′′+𝑦^′′𝑖. (2.8)
Hence, 𝑑𝑦=𝑦𝑑𝑥𝑥^′′=−𝑝−𝑞,𝑑𝑝+𝑞₂𝑦𝑑𝑥²=𝑦^′′𝑥^′−𝑦^′𝑥^′′𝑥^′2=2𝑟⁻⁴−𝑟⁴𝑝²+𝑞²𝑥^′2𝑟⁻²−𝑟²cos²2𝜃+𝑟⁻²+𝑟²²sin2𝜃>0. (2.9)
Therefore, the image of 𝐿(𝑟)={𝑟𝑒^𝑖𝜃;0<𝜃<𝜋/4} is a descending convex curve. By the symmetry of square, 𝐺₁({|𝑧|=𝑟}) is a smooth curve, and 𝐺₁({|𝑧|<𝑟<1}) is a convex symmetric figure in the square 𝑄.
Then, we obtain Lemma 2.2.

Figure 1 

Lemma 2.3. Suppose that mappings 𝐶(𝑧)(),𝐷(≜ℎ+𝑖𝑧−ℎ)(whereℎisdefinedinLemma2.2𝑧,𝑧)≜𝑥ln𝑅ℎ+𝑖𝑦𝜀ℎ=log𝑅+𝜀2ℎ𝑧+log𝑅−𝜀2ℎ𝐻(𝑧)𝑧,≜𝑒^𝑧,𝐺₂(𝑧)≜𝐶⁻¹∘𝐷⁻¹∘𝐻⁻¹(𝑧). (2.10) Then, the mapping 𝐺₂ maps the region 𝐸≜{1/𝑅<|𝑧|<𝑅}⋂{|arg𝑧|⩽𝜀} into the square 𝑄, and 𝐺₁⁻¹∘𝐺₂(𝐸^∗)⊂{|𝑧|<𝑟}, where 𝐸^∗≜{1/𝑅^∗<|𝑧|<𝑅^∗}⋂{|arg𝑧|⩽𝜀^∗}, 𝑅^∗∈(2,𝑅), 𝜀^∗∈(0,𝜀), and 0<1−𝑟<min𝜀−𝜀^∗22𝜋𝜀²,𝜀−𝜀^∗ln𝑅−ln𝑅^∗.4𝜋𝜀ln𝑅 (2.11)

Proof. The conclusion is equivalent to 𝐺₂(𝐸^∗)⊂𝐺₁({|𝑧|<𝑟}). We prove the lemma by two cases.
Case I. When 𝜀^∗𝜀⩾ln𝑅^∗,ln𝑅 (2.12) then 0⩽arg𝐺₂ℎ−ℎ𝜀^∗𝜀+𝑖(ℎ−ℎln𝑅^∗)ℎ/ln𝑅=arg𝜀−𝜀^∗𝜀+𝑖ln𝑅−ln𝑅^∗⩽𝜋ln𝑅4. (2.13)𝐺₁({|𝑧|<𝑟<1}) is a convex symmetric figure if there exists a 𝜃₀∈(0,𝜋/4], such that Re𝐺₁𝑟𝑒^𝑖𝜃0<ℎ𝜀−𝜀^∗𝜀,Im𝐺₁𝑟𝑒^𝑖𝜃0<ℎln𝑅−ln𝑅^∗,ln𝑅 (2.14) then Lemma 2.3 holds, see Figure 2.
In fact, for any 𝑟∈(0,1), 𝜃∈(0,𝜋/4], 𝐺₁𝑟𝑒^𝑖𝜃=−^𝑧₁(1+𝑖)𝑑𝑧√1−𝑧⁴=^𝑒𝑖𝜃₁(1+𝑖)𝑑𝑧√1−𝑧⁴+^{𝑟𝑒𝑖𝜃}_𝑒𝑖𝜃(1+𝑖)𝑑𝑧√1−𝑧⁴≜𝛼+𝛽, (2.15) where 𝛼=^𝑒𝑖𝜃₁(1+𝑖)𝑑𝑧√1−𝑧⁴=^𝜃₀−(1+𝑖)𝑖𝑒^𝑖𝜃1−𝑒^𝑖4𝜃=𝑑𝜃_𝜃0𝑑𝜃√<√sin2𝜃𝜋2^𝜃₀𝑑𝜃√𝜃=√𝜋𝜃,𝛽=^{𝑟𝑒𝑖𝜃}_𝑒𝑖𝜃(1+𝑖)𝑑𝑧√1−𝑧⁴=(1+𝑖)¹_𝑟𝑑𝑟1−𝑟⁴cos2𝜃−𝑖1+𝑟⁴≜()sin2𝜃1+𝑖¹_𝑟𝑑𝑟√,𝑎−𝑏𝑖 (2.16) where 𝑎=1−𝑟⁴cos2𝜃<1,𝑏=1+𝑟⁴𝑎sin2𝜃,₂+𝑏²=1+𝑟⁸+2𝑟⁴sin²2𝜃−cos²=2𝜃1−𝑟⁴+4𝑟⁴sin²2𝜃>4𝑟⁴sin2𝜃. (2.17)
By Lemma 2.1, we have 𝛽=¹_𝑟(1+𝑖)𝑝+𝑖𝑞𝑑𝑟=¹_𝑟𝑝−𝑞+𝑝+𝑞𝑖𝑑𝑟≜𝜉+𝑖𝜂, (2.18) where 𝜉=¹_𝑟𝑝−𝑞𝑑𝑟=¹_𝑟𝑝²−𝑞²𝑝+𝑞𝑑𝑟<¹_𝑟𝑝²+𝑞²𝑝<𝑑𝑟_1𝑟√2𝑑𝑟𝑎²+𝑏²<¹_𝑟𝑑𝑟𝑟²√<2sin2𝜃1−𝑟2𝑟√<sin2𝜃1−𝑟2𝑟𝜋𝜃,𝜂=¹_𝑟𝑝+𝑞𝑑𝑟<2¹_𝑟𝑝𝑑𝑟<2¹_𝑟𝑑𝑟₄𝑎²+𝑏²<1−𝑟𝑟𝜋.2𝜃 (2.19)
Let 𝜃₀=𝜀−𝜀^∗24𝜋𝜀²<𝜋4. (2.20)
Combing (3.1) (note that by (3.1), we have 𝑟>√2/2), Re𝐺₁𝑟𝑒^𝑖𝜃0=𝛼+𝜉<𝜋𝜃₀+1−𝑟2𝑟𝜋𝜃₀<ℎ𝜀−𝜀^∗𝜀,Im𝐺₁𝑟𝑒^𝑖𝜃0=𝜂<1−𝑟𝑟𝜋<12𝜃2ln𝑅−ln𝑅^∗ln𝑅<ℎln𝑅−ln𝑅^∗.ln𝑅 (2.21)
Therefore, a vertex of 𝐺₂(𝐸^∗)(ℎ−𝜀^∗ℎ/𝜀,ℎ−ℎln𝑅^∗/ln𝑅)∈𝐺₁({|𝑧|<𝑟}). By Lemma 2.2, 𝐺₂(𝐸^∗)⊂𝐺₁({|𝑧|<𝑟}).
Case II. When 𝜀^∗𝜀≤ln𝑅^∗,ln𝑅 (2.22) since 𝐺₁({|𝑧|<𝑟<1}) is a convex symmetric figure, we also have Lemma 2.3, see Figure 3.

Figure 2 

Figure 3 

For the convenience of readers, we prove the following lemma again, it can be found in [9].

Lemma 2.4. (1) Let 𝐺(𝑧)=𝐺₂⁻¹∘𝐺₁(𝑧)𝑒^𝑖𝜙, then 𝐺_𝑧+𝐺_𝑧⩽ln𝑅𝜀𝐺_𝑧−𝐺_𝑧. (2.23)
(2) Put 𝑠(𝑥,𝑦)=Re𝐺, 𝑡(𝑥,𝑦)=Im𝐺, then 𝑠²_𝑥+𝑠²_𝑦+𝑡²_𝑥+𝑡²_𝑦⩽ln𝑅𝜀𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥. (2.24)

Proof. (1) Since 𝑓≜𝐻⁻¹∘𝐺₁(𝑧)𝑒^𝑖𝜙 and 𝐶⁻¹ are holomorphic functions, then 𝑓_𝑧=𝑓_𝑧=𝐶⁻¹_𝐷−1=0,𝑓_𝑧=𝑓_𝑧. (2.25)
For 𝐷(𝑧)=𝑥ln𝑅ℎ+𝑖𝑦𝜀ℎ (2.26) then 𝐷⁻¹𝑓,𝑓=𝑥ℎln𝑅+𝑖𝑦ℎ𝜀=ℎ𝑓+𝑓+ℎ2ln𝑅𝑓−𝑓𝜀. (2.27) Hence, 𝐷max^𝑓−1+𝐷⁻¹_𝑓𝐷_𝑓⁻¹−𝐷⁻¹_𝑓=|ln𝑅+𝜀|+|ln𝑅−𝜀|||−||=ln𝑅+𝜀ln𝑅−𝜀ln𝑅𝜀. (2.28)
Thus, 𝐺_𝑧+𝐺_𝑧=𝐶⁻¹_𝐷−1𝐷_𝑓⁻¹𝑓_𝑧+𝐶⁻¹_𝐷−1𝐷⁻¹_𝑓𝑓_𝑧⩽𝐶⁻¹_𝐷−1𝑓_𝑧ln𝑅𝜀𝐷_𝑓⁻¹−𝐷⁻¹_𝑓=ln𝑅𝜀𝐶⁻¹_𝐷−1𝐷_𝑓⁻¹𝑓_𝑧−𝐶⁻¹_𝐷−1𝐷⁻¹_𝑓𝑓_𝑧. (2.29)
(2) By 𝑠_𝑥=𝑠_𝑧+𝑠_𝑧𝑠_𝑦=𝑠_𝑧−𝑖𝑠_𝑧, (2.30) we have 𝑠_𝑧=𝑠_𝑥−𝑖𝑠_𝑦1−𝑖,𝑠_𝑧=𝑠_𝑦−𝑖𝑠_𝑥.1−𝑖 (2.31) Similarly, 𝑡_𝑧=𝑡_𝑥−𝑖𝑡_𝑦1−𝑖,𝑡_𝑧=𝑡_𝑦−𝑖𝑡_𝑥.1−𝑖 (2.32)
Then, 𝐺_𝑧²=12𝑡_𝑦+𝑠_𝑥²+𝑡_𝑥−𝑠_𝑦²,𝐺_𝑧²=12𝑡_𝑥+𝑠_𝑦²+𝑡_𝑦−𝑠_𝑥². (2.33)
Therefore, 𝑠²_𝑥+𝑠²_𝑦+𝑡²_𝑥+𝑡²_𝑦=𝐺_𝑧²+𝐺_𝑧²⩽𝐺_𝑧+𝐺_𝑧²by(2.23)⩽ln𝑅𝜀𝐺_𝑧²−𝐺_𝑧²⩽ln𝑅𝜀𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥. (2.34)

Lemma 2.5 (see [10]). Let 𝐹 be a connected covering surface on 𝐹₀, 𝐹₀ is bounded by 𝑞 different points with radius 𝛿∈(0,1/2), then max0,𝜌(𝐹)⩾𝑞−2𝑆−2²⁵𝜋¹¹𝛿⁻¹⁹𝐿, (2.35) where 𝐿 is the length of 𝐹 and 𝜌(𝐹) is Euler characteristic of 𝐹, |𝐹| is the area of 𝐹 and |𝐹|𝑆=𝐹₀. (2.36)

Lemma 2.6 (see [10]). Let 𝑉 be a sphere with radius 1/2, 𝐹₀ be bounded by 𝑞 different points with radius 𝛿∈(0,1/2) and 𝐹_𝑟=𝑊∘𝐺(𝐹₀) then 𝐹𝑆=^𝑟𝐹₀=1𝜋^𝑣_𝑘=1^𝑟₀₀^2𝜋𝑤^′_𝑘²𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥𝑟1+𝑤_𝑘∘𝐺²²𝑑𝑟𝑑𝜃, (2.37) where 𝑠(𝑥,𝑦)=Re𝐺, 𝑡(𝑥,𝑦)=Im𝐺.

Proof. Suppose that 𝑤_𝑘=𝑢+𝑖𝑣. Then 𝐹_𝑟=^𝑣_𝑘=1𝐺(|𝑧_|<𝑟)11+𝑤_𝑘²²𝑑𝑢𝑑𝑣, (2.38) where 𝑑𝑢=𝑢_𝑠𝑠_𝑥+𝑢_𝑡𝑡_𝑥𝑑𝑥+𝑢_𝑠𝑠_𝑦+𝑢_𝑡𝑡_𝑦𝑣𝑑𝑦,𝑑𝑣=_𝑠𝑠_𝑥+𝑣_𝑡𝑡_𝑥𝑑𝑥+𝑣_𝑠𝑠_𝑦+𝑣_𝑡𝑡_𝑦𝑑𝑦. (2.39)
Hence, by the Jacobian determinant, we have 𝑑𝑢𝑑𝑣=𝑢²_𝑠+𝑣²_𝑠𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥𝐹𝑑𝑥𝑑𝑦,_𝑟=^𝑣_𝑘=1|𝑧_|<𝑟𝑢²_𝑠+𝑣²_𝑠𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥1+𝑤_𝑘∘𝐺²²=𝑑𝑥𝑑𝑦_{𝑣𝑘=1𝑟002𝜋}𝑤^′_𝑘²𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥𝑟1+𝑤_𝑘∘𝐺²²𝑑𝑟𝑑𝜃. (2.40)
By |𝐹₀|=𝜋, we have Lemma 2.6.

3. Proof of Theorem 1.1

Proof. Set 𝐺(𝑧)=𝐺₂⁻¹∘𝐺₁(𝑧)𝑒^𝑖𝜙. It conforms the unit disc 𝑈={|𝑧|<1} to the sector 𝐸={1/𝑅<|𝑊|<𝑅}⋂{|arg𝑧−𝜙|<𝜀} and the interior of 𝑈^∗={|𝑧|<𝑟} to 𝐸^∗={1/𝑅^∗<|𝑊|<𝑅^∗}⋂{|arg𝑧−𝜙|<𝜀^∗}, where 2<𝑅^∗<𝑅, 0<𝜀^∗<𝜀, and 0<1−𝑟<min𝜀−𝜀^∗22𝜋𝜀²,𝜀−𝜀^∗ln𝑅−ln𝑅^∗.4𝜋𝜀ln𝑅 (3.1) Hence 𝑊∘𝐺 conforms {|𝑧|<𝑟} to the sphere 𝑉.
Put 𝐷_𝑟={|𝑧|<𝑟}. Then by M. Hurwite Formula, we have 𝜌𝐷_𝑟=𝑛𝑟,𝑅_𝑧−𝑣. (3.2) Put 𝐷_𝑟=𝐷_𝑟−{𝑧∣∏^𝑞_𝑗=1(𝑤∘𝐺(𝑧)−𝑎_𝑗)=0} and 𝐹_𝑟=𝑊∘𝐺(𝐹₀). Then 𝜌𝐹_𝑟=𝜌𝐷_𝑟=𝑛𝑟,𝑅_𝑧−𝑣+^𝑞_𝑗=1𝑛𝑟,𝑊∘𝐺=𝑎_𝑗⩽𝑛1,𝑅_𝑧−𝑣+^𝑞_𝑗=1𝑛1,𝑊∘𝐺=𝑎_𝑗≜𝑁. (3.3) By Lemma 2.5, it follows that 𝑁⩾𝜌𝐹_𝑟⩾𝑞−2𝑆(𝑟,𝑊∘𝐺)−2²⁷𝜋¹¹𝛿⁻¹⁹𝐿(𝑟). (3.4) Now we will prove 𝐿²(𝑟)⩽2⁴𝑣𝜋𝑟ln𝑅𝜀𝑑𝑆(𝑟,𝑊∘𝐺).𝑑𝑟 (3.5) For any 𝑟∈(0,1), 𝑘=1,2,…,𝑣 and 𝜀>0, we have 𝑤_𝑘∘𝐺𝑟𝑒^𝑖𝜃−𝑤_𝑘∘𝐺^𝑗<𝜀, (3.6) where 𝜃_𝑗=𝑗𝜋/𝑛(𝑗=1,2,…,2𝑛), 𝜃∈[𝜃_𝑗−1,𝜃_𝑗], |𝑤_𝑘∘𝐺^𝑗|=min{|𝑤_𝑘∘𝐺(𝑟𝑒^𝑖𝜃)|,𝜃_𝑗−1⩽𝜃⩽𝜃_𝑗}.
By (3.6), for any 𝜃∈[𝜃_𝑗−1,𝜃_𝑗], we have 𝑤_𝑘∘𝐺𝑟𝑒^𝑖𝜃2⩽𝑤_𝑘∘𝐺^𝑗2+2𝜀𝜀+𝑤_𝑘∘𝐺^𝑗. (3.7) Therefore 1+𝑤_𝑘∘𝐺𝑟𝑒^𝑖𝜃21+𝑤_𝑘∘𝐺^𝑗2⩽1+2𝜀²1+𝑤_𝑘∘𝐺^𝑗2+2𝜀𝑤_𝑘∘𝐺^𝑗1+𝑤_𝑘∘𝐺^𝑗2⩽√2. (3.8) Put 𝑤_𝑘=𝑢_𝑘(𝑠,𝑡)+𝑖𝑣_𝑘(𝑠,𝑡), 𝐺=𝑠(𝑟,𝜃)+𝑖𝑡(𝑟,𝜃). Hence 𝐿_𝑘(𝑟)≜lim_𝑛→∞^2𝑛_𝑗=1𝑤_𝑘∘𝐺𝑟𝑒^𝑖𝜃𝑗−𝑤_𝑘∘𝐺𝑟𝑒^{𝑖𝜃𝑗−1}1+𝑤_𝑘∘𝐺𝑟𝑒^{𝑖𝜃𝑗2}1+𝑤_𝑘∘𝐺𝑟𝑒^{𝑖𝜃𝑗−12}⩽lim_𝑛→∞^2𝑛_𝑗=1∫^𝜃𝑗_𝜃𝑗−1𝑢_𝑘𝑠𝑠_𝜃+𝑢_𝑡𝑠𝑡_𝜃𝑑𝜃²+∫^𝜃𝑗_𝜃𝑗−1𝑣_𝑘𝑠𝑠_𝜃+𝑣_𝑘𝑡𝑡_𝜃𝑑𝜃^21/21+𝑤_𝑘∘𝐺^𝑗2⩽√2lim_𝑛→∞^2𝑛_𝑗=1^𝜃𝑗_𝜃𝑗−1𝑢_𝑘𝑠𝑠_𝜃+𝑢_𝑘𝑡𝑡_𝜃²+𝑣_𝑘𝑠𝑠_𝜃+𝑣_𝑘𝑡𝑡_𝜃^21/2𝑑𝜃1+𝑤_𝑘∘𝐺^𝑗_𝑘²=2₀^2𝜋𝑤^′_𝑘𝑠²_𝜃+𝑡²_𝜃^1/2𝑑𝜃1+𝑤_𝑘∘𝐺_𝑘². (3.9)
By 𝑠_𝜃=−𝑠_𝑥𝑟sin𝜃+𝑠_𝑦𝑡𝑟cos𝜃,_𝜃=−𝑡_𝑥𝑟sin𝜃+𝑡−𝑡_𝑦𝑟sin𝜃, (3.10) where 𝑥=−𝑟cos𝜃,𝑦=𝑟sin𝜃, (3.11) we obtain 𝑠²_𝜃+𝑡²_𝜃⩽2𝑟²𝑠²_𝑥+𝑠²_𝑦+𝑡²_𝑥+𝑡²_𝑦,𝐿²(𝑟)=^𝑣_𝑘=1𝐿_𝑘(𝑟)²⩽8^𝑣_𝑘=10^2𝜋𝑤^′_𝑘𝑠²_𝑥+𝑠²_𝑦+𝑡²_𝑥+𝑡²_𝑦^1/2𝑟1+𝑤_𝑘∘𝐺²𝑑𝜃²by(2.24)⩽8ln𝑅𝜀^𝑣_𝑘=10^2𝜋𝑤^′_𝑘𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥^1/2𝑟1+𝑤_𝑘∘𝐺²𝑑𝜃²Cauchyinequality⩽8𝑣ln𝑅𝜀^𝑣_𝑘=10^2𝜋𝑤^′_𝑘𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥^1/2𝑟1+𝑤_𝑘∘𝐺²𝑑𝜃²Schwarzinequality⩽8𝑣ln𝑅𝜀^𝑣_𝑘=10^2𝜋𝑤^′_𝑘²𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥𝑟1+𝑤_𝑘∘𝐺²²𝑑𝜃₀^2𝜋𝑟𝑑𝜃⩽16𝑣𝜋𝑟ln𝑅𝜀^𝑣_𝑘=10^2𝜋𝑤^′_𝑘²𝑠_𝑥𝑡_𝑦−𝑠_𝑦𝑡_𝑥𝑟1+𝑤_𝑘∘𝐺²²𝑑𝜃byLemma2.6⩽16𝑣𝜋𝑟ln𝑅𝜀𝑑𝑆(𝑟,𝑊∘𝐺).𝑑𝑟 (3.12)
(1) If forall𝑟^′∈(𝑟,1)𝑆()⩾𝑁𝑟,𝑊∘𝐺,𝑞−2 (3.13) then by (3.4) 𝑆𝑟^′−𝑁,𝑊∘𝐺𝑞−2²⩽2⁵⁰𝜋²²𝑞−2²𝛿³⁸𝐿𝑟^′()⩽2by3.5⁵⁴𝑣𝜋²³𝑟^′ln𝑅𝑞−2²𝛿³⁸𝜀𝑑𝑆𝑟^′,𝑊∘𝐺𝑑𝑟^′, (3.14) that is, 𝑑𝑟^′⩽2⁵⁴𝑣𝜋²³ln𝑅𝑞−2²𝛿³⁸𝜀𝑑𝑆𝑟^′,𝑊∘𝐺𝑆𝑟^′,𝑊∘𝐺−𝑁/𝑞−2². (3.15) Hence, 1−𝑟=¹_𝑟𝑑𝑟^′⩽2⁵⁴𝑣𝜋²³ln𝑅𝑞−2²𝛿³⁸𝜀^1𝑟𝑑𝑆𝑟^′,𝑊∘𝐺𝑆𝑟^′,𝑊∘𝐺−𝑁/𝑞−2²⩽2⁵⁴𝑣𝜋²³ln𝑅𝑞−2𝛿³⁸𝜀1𝑆()−1𝑟,𝑊∘𝐺−𝑁/𝑞−2𝑆()<2𝑅,𝑊∘𝐺−𝑁/𝑞−2₅₄𝑣𝜋²³ln𝑅𝛿³⁸𝜀1𝑆().𝑞−2𝑟,𝑊∘𝐺−𝑁 (3.16) Therefore 𝑆()⩽2𝑞−2𝑟,𝑊∘𝐺⁵⁴𝑣𝜋²³ln𝑅𝛿³⁸𝜀()𝑅1−𝑟+𝑁⩽𝑛1,_𝑧+^𝑞_𝑗=1𝑛1,𝑊∘𝐺=𝑎_𝑗+2⁵⁴𝑣𝜋²³ln𝑅𝛿³⁸𝜀().1−𝑟 (3.17)
(2) If there is a 𝑟^′∈(𝑟,1), such that 𝑞−2𝑆𝑟^′,𝑊∘𝐺−𝑁<0, (3.18) then 𝑞−2𝑆(𝑟,𝑊∘𝐺)<𝑞−2𝑆𝑟^′,𝑊∘𝐺<𝑁, (3.19) Equation (3.17) holds.
By (3.17) and Lemma 2.3, we have 𝑞−2𝑆Ω𝜙−𝜀^∗,𝜙+𝜀^∗,𝑅^∗,𝑊−𝑞−2𝑆Ω𝜙−𝜀^∗,𝜙+𝜀^∗,1𝑅^∗⩽𝑆()()Ω𝑅,𝑊𝑞−2𝑟,𝑊∘𝐺by3.1⩽𝑛𝜙−𝜀,𝜙+𝜀,𝑅,_𝑧+^𝑞_𝑗=1𝑛Ω𝜙−𝜀,𝜙+𝜀,𝑅,𝑊=𝑎_𝑗+2⁵⁶𝑣𝜋²⁴ln𝑅𝛿³⁸(𝜀−𝜀^∗)(ln𝑅−ln𝑅^∗). (3.20)

4. Proof of Theorem 1.2

Proof. By the hypothesis of Theorem 1.2, there exists an increasing sequence 𝑅_𝑛 (𝑅_𝑛→∞, when 𝑛→∞), such that lim_𝑛→∞𝑇𝑅_𝑛,𝑊ln²𝑅_𝑛=+∞. (4.1)
Then, there exist some 𝜙₀∈[0,2𝜋], such that for arbitrary 𝜀∈(0,𝜙₀), lim_𝑛→∞𝑇𝑅_𝑛,𝜙₀−𝜀,𝜙₀+𝜀,𝑊ln²𝑅_𝑛=+∞ (4.2) holds. We claim that arg𝑧=𝜙₀ is the Nevanlinna direction.
Otherwise, for a positive number 𝜀₀, there exist some 𝑎₁,𝑎₂,…,𝑎_𝑞(𝑞≥3), such that ^𝑞_𝑗=1𝛿𝑎_𝑗,𝜙₀>2+3𝜀₀. (4.3) By the definition of 𝛿(𝑎_𝑗,𝜙₀), we have lim_𝜀→0+lim_𝑅→∞∑^𝑞_𝑗=1𝑁Ω𝜙₀−𝜀,𝜙₀+𝜀,𝑅,𝑊=𝑎_𝑗𝑇Ω𝜙₀−𝜀,𝜙₀+𝜀,𝑅,𝑊<𝑞−2−3𝜀₀. (4.4) There exists 𝜀₁>0, such that for any 𝜀∈(0,𝜀₁), lim_𝑅→∞∑^𝑞_𝑗=1𝑁Ω𝜙₀−𝜀,𝜙₀+𝜀,𝑅,𝑊=𝑎_𝑗𝑇Ω𝜙₀−𝜀,𝜙₀+𝜀,𝑅,𝑊<𝑞−2−2𝜀₀. (4.5)
Hence, for {𝑅_𝑛} defined earlier, when 𝑛 is sufficiently large, ^𝑞_𝑗=1𝑁Ω𝜙₀−𝜀,𝜙₀+𝜀,𝑅_𝑛,𝑊=𝑎_𝑗<𝑞−2−𝜀₀𝑇Ω𝜙₀−𝜀,𝜙₀+𝜀,𝑅_𝑛,𝑊. (4.6) By Theorem 1.1, we have 𝑆Ω𝜀𝑞−2𝜙−2𝜀,𝜙+2−𝑆Ω𝜀,𝑅,𝑊𝑞−2𝜙−2𝜀,𝜙+2⩽,1,𝑊_𝑞𝑗=1𝑛Ω𝜙−𝜀,𝜙+𝜀,2𝑅,𝑊=𝑎_𝑗+2⁵²𝜋²⁴ln2𝑅𝛿³⁸.𝜀ln2 (4.7) Hence, 𝑇Ω𝜀𝑞−2𝜙−2𝜀,𝜙+2,𝑅_𝑛⩽,𝑊_𝑞𝑗=1𝑁Ω𝜙−𝜀,𝜙+𝜀,𝑅_𝑛,𝑊=𝑎_𝑗+2⁵⁴𝜋²⁴ln²2𝑅_𝑛𝛿³⁸(𝜀−𝜀^∗)(1)<ln2+𝑂𝑞−2−𝜀₀+𝑂(1)ln²𝑅_𝑛. (4.8) Hence, lim_𝑛→∞𝑇Ω𝜙−𝜀/2,𝜙+𝜀/2,𝑅_𝑛,𝑊ln²𝑅_𝑛<𝑂(1), (4.9) which contradicts (4.2). Therefore, Theorem 1.2 holds.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (no. 11101096), Guangdong Natural Science Foundation (no. S2012010010376), and the Startup Foundation for Doctors of Guangdong University of Technology (no. 083063).