Mathematical Sciences: Value Distribution of Complex Functions

在整函数和亚纯函数的数学理论中，这项工作的两个主要目的是给出亚纯函数的和的精确界和分析整函数的最小模。亚纯函数的亏量是函数对给定复数的亲和力的一种度量。一般说来，亏量是无穷的，也就是说，亚纯函数经常无穷地达到最复杂的值。对于有限级函数，已知缺陷之和不大于两个。不足两个的数量应该根据函数的顺序来衡量。猜想的上界(许多年前由阿尔伯特·埃德雷提出)已被证明对大子函数类有效。有相当多的证据表明，这一基本估计的完全解决方案可能近在咫尺。这个项目的第二个重点是比较指定阶次的整(或次调和)函数的最大和最小模数。将模数与基础变量的大值进行比较。当函数的阶数小于1时，Wiman-Valron的经典结果给出了一个尖锐的下界。对于等于或大于1的顺序，几乎不知道确切的信息。新的、相关的结果可能会提供与此问题相关的信息。研究的重点将是将零点的数量和分布与极端情况联系起来。这项工作有望对复微分方程和拟共形映射具有辅助应用。

Two of the main objectives of this work in the mathematical theory of entire and meromorphic functions are to give sharp bounds for the sum of Nevanlinna deficiencies of a meromorphic function and to analyze the minimum modulus of an entire function. The deficiency of a meromorphic function is a measure of the function's affinity for a given complex number. Generally the deficiency is infinite, that is, meromorphic functions achieve most complex values infinitely often. For functions of finite order the sum of the defects is known to be no greater than two. The amount this falls short of two should be measured in terms of the order of the function. The conjectured upper bounds (proposed many years ago by Albert Edrei) have been proved to be valid for large subclasses of functions. There is considerable evidence to suggest that a complete solution of this fundamental estimate may be near at hand. The second thrust of this project seeks to compare the maximum and minimum moduli of entire (or subharmonic) functions of prescribed order. The moduli are compared for large values of the underlying variable. When the order of the function is less than one, the classic result of Wiman-Valiron gives a sharp lower bound. For order equal to unity or above little exact information is known. New, related, results may provide information related to this question. The focus of the research will be on relating the number and distribution of zeros to the extremal cases. This work is expected to have ancillary applications to complex differential equations and quasiconformal mapping.