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