涉及差分算子的亚纯函数的Picard型定理与正规定则

The theory for value distribution and normal family has played an important role in the functions of one complex variable, and it has not only very important theoretical value, but also has a wide range of applications in complex dynamical systems, the distribution for the zeros of the solutions of complex differental equation and uniqueness theory. In recent years, many research has emerged in replacing the derivative of functions to the difference operator of functions, and also two fundamental theorems concerning difference operator for mermorphic functions have been proven. Through the study of this project, we hope to get the Picard-Hayman type of inequalitiy, theorem, normal criterion, and the applications in the properites of the solutions of complex difference equations.

亚纯函数的值分布与正规族理论是单复变函数论中的一个重要内容，不仅自身有着重要理论价值，而且在复解析动力系统，复微分方程解的零点分布理论，及唯一性理论中有着广泛的应用。近年来，该理论的研究中出现了将传统的涉及函数导数的结果改为涉及函数差分的结论，取得了重要的进展，相继得到了涉及差分算子的两个基本定理。通过本项目的研究，希望得到涉及差分算子的亚纯函数的Picard-Hayman型不等式，定理和相应的正规定则，并探索其在复差分方程解的性质中的应用。

我们按照计划研究了涉及c差分算子和q差分算子的Hayman不等式及其相对应的Picard-Hayman型定理, 得到了极点密指量N(r,f)为小函数的有穷级亚纯函数涉及c差分算子的Hayman定理. 同时, 利用正规定则研究了亚纯函数涉及q差分算子的Hayman定理, 得到了相应的结果, 该结果拓展了涉及差分算子值分布的研究工作, 使之前仅对有穷级函数成立的结果可以推广到无穷级的情况.