Abstract

The maximum of the modulus of a meromorphic function cannot be restricted from above by the Nevanlinna characteristic of this meromorphic function. But integrals from the logarithm of the module of a meromorphic function allow similar restrictions from above. This is illustrated by one of the important theorems of Rolf Nevanlinna in the classical monograph by A. A. Goldberg and I. V. Ostrovskii on meromorphic functions, as well as by the Edrei–Fuchs Lemma on small arcs and its versions for small intervals in articles by A. F. Grishin, M. L. Sodin, T. I. Malyutina. Similar results for integrals of differences of subharmonic functions even with weights were recently obtained by B. N. Khabiblullin, L. A. Gabdrakhmanova. All these results are on integrals over subsets on a ray. In this article, we establish such results for integrals of the logarithm of the modulus of a meromorphic function and the difference of subharmonic functions over discs and planar small sets. Our estimates are uniform in the sense that the constants in these estimates are explicitly written out and do not depend on meromorphic functions and the difference of subharmonic functions provided that these functions has an integral normalization near zero.

中文翻译：

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具有亚纯函数的积分或圆盘和平面小集上的次谐波函数的差分

摘要

亚纯函数的模的最大值不能被这个亚纯函数的 Nevanlinna 特性从上面限制。但是来自亚纯函数模的对数的积分允许上述类似的限制。AA Goldberg 和 IV Ostrovskii 的经典专着中关于亚纯函数的 Rolf Nevanlinna 的一个重要定理，以及 AF Grishin 文章中关于小弧的 Edrei-Fuchs 引理及其版本的小区间版本，都说明了这一点， ML Sodin，TI Malyutina。最近由 BN Khabiblullin, LA Gabdrakhmanova 获得了即使具有权重的次谐波函数差异积分的类似结果。所有这些结果都是关于射线子集的积分。在本文中，我们为亚纯函数的模数的对数积分以及圆盘和平面小集上的次谐波函数的差异建立了这样的结果。我们的估计是统一的，因为这些估计中的常数被明确写出并且不依赖于亚纯函数和次谐波函数的差异，前提是这些函数具有接近零的积分归一化。