圆内亚纯函数的Julia型奇异点

SINGULAR POINTS OF JULIA TYPE OF MEROMORPHIC FUNCTIONS IN UNIT DISK

对单位圆内的亚纯函数提出了一种与正规族理论中著名的Marty定则相对应的奇异点—Marty点的概念 首先讨论了Marty点存在的条件 ,并由此证明了如下结论 :如果lim-r→ 1-T(r ,f)(log 11-r) 2=+∞ ,则存在点eiθ(0≤θ <2π) ,使得对任意有穷非零复数a ,任意正数ε和任意正整数n有limr→ 1-n(Ω(θ -ε ,θ +ε,r) ,fnf′ =a)

Yang Lo once posed a question: Is there a singular direction corresponding to every normal family? In this paper we discuss this question for meromorphic functions in the unit disk. To do this, we define a new kind of singular points which correspond to the famous Marty's criterion. Definition. Let f be a meromorphic function in unit disk and θ (∈[0,2π]) a real number. If sup z∈Ω(θ-ε,θ+ε;1)1-｜z｜ k+1 ｜f′(z)｜1+(1-｜z｜) 2k ｜f(z)｜ 2=+∞ , then e iθ is called a Marty's point of f. We not only give a good condition concerning the existence of Marty's points, but also prove that if e iθ is a Marty's point of f, then f f' can infinitely take an arbitrary finitely nonzero number in the angular domain Ω(θ-ε,θ+ε; 1) for any positive number and integer n. In the process, it is notable that we only use the first fundemental theorem with the former and the normal criterion of Marty with the latter.