Abstract

Let \(D\) be a domain in the complex plane and \(M\) be an extended real-valued function on \(D\). If \(f\) is a non-zero holomorphic function on \(D\) such that \(|f|\leq\exp M\), then it is natural to expect that there should be some upper boundedness for the distribution of the zeros of \(f\) expressed exclusively in terms of the function \(M\) and the geometry of the domain \(D\). We have investigated this question in detail in our previous works in the case when \(M\) is a subharmonic function and the domain \(D\) either is arbitrary or has a non-polar boundary. The answer was given in terms of constraints to the distribution of zeros of \(f\) from above via the Riesz measure of the subharmonic function \(M\). In this article, the function \(M\) is the difference of subharmonic functions, or a \(\delta\)-subharmonic function, and the upper constraints are given in terms of the Riesz charge of this \(\delta\)-subharmonic function \(M\). These results are also new to a certain extent for the subharmonic function \(M\). The case when the domain \(D\) coincides with the whole complex plane is considered separately. For the complex plane, it is possible to reach the criterion level of our results.

中文翻译：

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平面域中具有上约束的全纯函数零子集的充要条件

摘要

让\(D\)是复平面中的域，而\(M\)是\(D\)上的扩展实值函数。如果\(f\)是\(D\)上的非零全纯函数，使得\(|f|\leq\exp M\)，那么很自然地期望分布应该有一些上限\(f\)的零点仅用函数\(M\)和域\(D\)的几何表示。我们在之前的工作中详细研究了这个问题，当\(M\)是一个次谐波函数并且域\(D\)要么是任意的，要么具有非极性边界。答案是通过对次谐波函数\(M\)的 Riesz 测度从上方对\(f\)的零点分布的约束给出的。在本文中，函数\(M\)是次谐波函数的差值，或一个\(\delta\) -次谐波函数，并且根据该\(\delta\)的Riesz 电荷给出了上限-次谐波函数\(M\)。这些结果在一定程度上对于次谐波函数\(M\)也是新的。当域\(D\)与整个复平面重合是分开考虑的。对于复平面，有可能达到我们结果的标准水平。