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

and Applied Analysis 3 is called type function of w z or T r,w such that ρ 1/ 1 − r is nondecreasing, piecewise continuous, and differentiable, and lim r→ 1− ρ ( 1 1 − r ) ρ, lim r→ 1− U k/ 1 − r U 1/ 1 − r k ρ (k is any given positive constant ) , lim r→ 1− T r,w U 1/ 1 − r 1, lim r→ 1− 1/ 1 − r ρ−ε U 1/ 1 − r 0, 0 < ε < ρ. 1.8 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 Γn : {|z − zn| < rnσn}, n 1, 2, . . . , 1.9 where zn rnen , lim n→∞ rn 1, σn > 0, lim n→∞ σn 0. 1.10 Such that for each α n Γn ∩ , w α ≥ 1 1 − rn ρ 1−εn , 1.11 except for those complex numbers contained in the union of 2ν spherical discs each with radius 1 − rn , 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 . 4 Abstract and Applied Analysis 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


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 equation where A ν z , . . . , A 0 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. Denote 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. Let where | z| r is the boundary of | z| ≤ r. The characteristic function of w z is defined by In view of 4 , we have The order of algebroid function w z is defined by 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. Write 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 Abstract and Applied Analysis 3 is called type function of w z or T r, w such that ρ 1/ 1 − r is nondecreasing, piecewise continuous, and differentiable, and 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 where z n r n e iθ n , lim n → ∞ r n 1, σ n > 0, lim n → ∞ σ n 0.

1.10
Such that for each α except for those complex numbers contained in the union of 2ν spherical discs each with radius 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−ε n . 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 .  an integer and a 1 , a 2 , a 3 , . . . , 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

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

2.2
where x stands for the inter part of x. Then, among p, q, there exists at least one pair p 0 , q 0 , such that 1 − a p 0 > R, and in Ω p 0 q 0 , except for those complex numbers contained in the union of 2ν spherical discs each with radius δ a p 0 ρ/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 / log a 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 j p, q 2ν 1 j 1 .

2.4
For any p, q mentioned above, we have n l Ω pq , w α j < 1 a p ρ 1−ε . 2.5 For any r > R, let T log 1 − r / log a , then we have 1 − a T ≤ r < 1 − a T 1 .

2.7
Thus there exists t 0 , j 0 which are related to T . We can assume t 0 0, j 0 0, such that

2.9
Then we have Since {Ω pq } p,q covers T −1 p 1 L p0 and m−1 q 0 q0 twice at most. We obtain 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 Ω 0 pq is contained in the disc |ζ| < η <1 . Since 6 Abstract and Applied Analysis all Ω pq , Ω 0 pq are similar, C is independent of p, q. Since S is conformally invariant, in view of Lemma 2.1, we obtain

2.12
For sufficiently large integer T log 1 − r / log a , r ∈ 1 − a T , 1 − a T 1 . Thus we get

2.13
where C is a constant.
Abstract and Applied Analysis 7

2.16
Next, we deduce the following:

2.18
Dividing both sides of 2.13 by νt and integrating it from 1 − a T to r, we have Note that T is fixed, we see that T 1 − a T −1 , w is a finite constant. Hence,

2.21
In view of 3 , we know that N r, R z ≤ 2 ν − 1 T r, w O 1 .

2.22
We obtain

2.23
where C is a constant.
Abstract and Applied Analysis 9 Dividing both sides of the above inequality by U 1/ 1 − r 1/ 1 − r ρ 1/ 1−r , we have