涉及分担小函数正规定则的研究

Research on the Normal Rule of Sharing Small Functions

摘要: Bloch曾经提出：相应于每一个Picad型定理，必有一个相应的正规定则。这个原理对正规族理论的发展起到了重要的作用。在此基础上，利用亚纯函数值分布理论、线性代数理论及第二基本定理研究方法，研究了分担小函数的正规定则。得到如下结论：设F是D⊂ℂ上的一族亚纯函数，a₁(z)，a₂(z)，a₃(z)在D上全纯，且满足。若对∀f∈F，有f(z)=a_i(z)⇔f^'(z)=a_i(z)，则F在D上正规。该结论拓展了思路，将小函数和正规族以及分担值结合在一块，对后续的研究提供了思维和方法。

Abstract: Bloch once proposed that there must be a corresponding normal rule for every Picad type theorem. This principle plays an important role in the development of normal family theory. On this basis, using the meromorphic function value distribution theory, linear algebra theory and the second basic theorem research method, the normal rule of sharing small functions is studied. The following conclusions are obtained: Let F is a family of meromorphic functions of a domain D⊂ℂ, a₁(z)，a₂(z)，a₃(z) is holomorphic on D, and they satisfied . Assume the condition hold for every f∈F, that f(z)=a_i(z)⇔f^'(z)=a_i(z), then F is normal on D.