圆环内亚纯函数与向量值亚纯函数的Nevanlinna理论

The main purpose of this program is to study the Nevanlinna theory for the meromorphic functions on annuli and vector-valued meromorphic functions of infinite dimension, and to study the value distribution of solutions of differential equation in complex domain by adopting Nevanlinna theory. The main contents of this program are as follows:.1. studying the exceptional values, deficiency, relative deficiency of meromorphic functions and their derivatives on annuli and the Nevanlinna characteristic functions of meromorphic functions and their derivatives on annuli with maximum deficiency sum;.2. studying the exceptional values, deficiency and relative deficiency for an E-valued meromorphic function from the complex plane to an infinite dimensional complex Banach space E and the Nevanlinna characteristic function of E-valued meromorphic functions and their derivatives with maximum deficiency sum;.3. studying the existence and the value distribution of quasimeromorphic function solutions for the complex differential equations, focusing on establishing the equivalence of the zero accumulation ray and radial growth order of the solutions of high order differential equations with entire coefficients of infinite order..The above-mentioned three new aspects deal with problems that researchers have recently paid attention to. Any solution of these problems is of theoretical significance to this subject and will enrich the related subjects.

本项目主要研究圆环内亚纯函数与无限维向量值亚纯函数的Nevanlinna理论和运用Nevanlinna理论研究复域微分方程解的值分布。主要研究内容如下：.1. 研究圆环内亚纯函数及其导函数的Borel例外值、亏量、相对亏量和具有最大亏量和的圆环内亚纯函数及其导函数的Nevanlinna特征函数；.2. 研究从复平面映射到无限维巴拿赫空间E的E-值亚纯函数及其导函数的Borel例外值、亏量、相对亏量和具有最大亏量和的E-值亚纯函数及其导函数的Nevanlinna特征函数；.3. 研究复域微分方程的拟亚纯函数解的存在性和解的值分布性质，着重研究无限级高阶整函数系数微分方程解的零点聚值线和径向增长级之间的等价关系。.上述3个方面的研究内容是富有新意的研究问题，是相关领域国内外最近才开始关注的问题，这些问题的任何进展对本学科有一定的理论意义，同时能推动相关学科的发展。

本项目研究了圆环内亚纯函数与向量值亚纯函数的Nevanlinna理论和并利用亚纯函数的Nevanlinna理论研究了复域微分方程解的值分布。主要完成的研究工作如下：.1. 建立了圆环内亚纯函数的Milloux不等式， 研究了圆环内亚纯函数的不动点、唯一性、亏量以及具有最大亏量和的圆环内亚纯函数及其导函数的Nevanlinna特征函数；.2. 研究了亚纯函数的差分的不动点和具有最大亏量和的亚纯函数及其导函数和差分算子的Nevanlinna特征函数；.3. 建立了E-值亚纯函数的Milloux不等式， 研究了E-值亚纯函数及其导函数的Borel例外值、亏量、相对亏量和具有最大亏量和的向量值亚纯函数E-值亚纯函数及其导函数的Nevanlinna特征函数；.4. 研究了复域微分方程的亚纯函数解的的值分布性质，验证了无限级高阶整函数系数微分方程的解的零点聚值线和径向增长级之间的等价关系。