亚纯函数在角域内的波莱耳方向

BOREL DIRECTIONS OF MEROMORPHIC FUNCTIONS IN AN ANGULAR DOMAIN

Suppose that f(z) is a meromorphic function of order λ(0<λ≤∞) and of lower order μ(0≤μ<∞) in the plane. Let ρ(μ≤ρ≤λ) be a finite positive number. B: arg z=θ₀(0≤θ₀ <2π) is called a Borel direction of order ρ of f(z), if for any complex number a, the equality holds, except at most for some a belonging to a set of linear measure zero. For the exceptional values a, we have ρ(θ₀, a)>ρ, except two possible values. With the above hypotheses on f(z), λ, μ and ρ, We have the following lemmas. Lemma 1. There exists a sequence of positive numbers (r_n) such that(?)=∞ and that Lemma 2. If f(z) has a deficient value a₀ with deficiency δ(a₀, f), then we have where (r_n) is the sequence defined in the Lemma 1 and when a_0=∞, we have to replace(?)by (?) in the left hand side of (*). Lemma 3. Suppose that B_1 : arg z =θ₁ and B₂ : arg z=θ₂ (0≤θ₁<θ₂<2π+θ₁) are two half straight lines from the origin and there are no Borel directions of order≥ρ(ρ>1/2) of f(z) in θ₁₀, the inequality holds as n is sufficiently large, where K₁ is a positive number not depending on n andεand when a₀=∞, it is necessary to replace we have θ₂-θ₁≤π/ρ. Theorem 1. Suppose that f(z) is a meromorphic function of order λ (1/2<λ≤+∞) and of lower order μ(0≤μ<+∞) in the plane. Let p be a number such that μ≤ρ≤λ and that 1/2<ρ<+∞If f~((k))(z) has p(1≤P<+∞) deficient values a_i (i=1,2,…,p) with deficiencies δ(a_i,f^(k)), then f(z) has a Borel direction of order ≥ρ in any angular domain, the magnitude of which is larger than It is convenient to consider Julia directions as Borel directions of order zero.Under this assumption, We have the following. Theorem 2. Suppose that f(z) is a meromorphic function of order λ and of finite lower order μ in the plane and that ρ(μ≤ρ≤λ) is a finite number. If p denotes the number of deficient values of f(z) and q denotes the number of Borel directions of order ≥p of f(z), then we have p≤q.