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The infinitesimal behavior of the sum of Cauchy kernels and its derivative at infinity

Vladimir Shemyakov

Abstract

In analysis, it's often useful to know the value of a function at infinity, this operation possesses pleasant properties. However, even when the limit does not exist, some intuitive considerations may suggest that the function still assumes a specific value at infinity in a certain sense. In Nevanlinna theory, all objects are studied on average, i.e., their integrals, hence the integral interpretation of the concept of convergence to a limit is beneficial for the theory of meromorphic functions. This is precisely the focus of this work, applied to sums of Cauchy kernels and their derivatives.

The infinitesimal behavior of the sum of Cauchy kernels and its derivative at infinity

Abstract

In analysis, it's often useful to know the value of a function at infinity, this operation possesses pleasant properties. However, even when the limit does not exist, some intuitive considerations may suggest that the function still assumes a specific value at infinity in a certain sense. In Nevanlinna theory, all objects are studied on average, i.e., their integrals, hence the integral interpretation of the concept of convergence to a limit is beneficial for the theory of meromorphic functions. This is precisely the focus of this work, applied to sums of Cauchy kernels and their derivatives.
Paper Structure (2 sections, 7 theorems, 61 equations)

This paper contains 2 sections, 7 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. The smallness of the derivative of the sum of Cauchy kernels at infinity

Key Result

Lemma 1

Let $f$ be analytic in the disk $|z|< r$ and $0< p< 1$. Then

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Lemma 1: V.I. Smirnov, Book_Gold_Ostr, 6, стр. 253, lemma 6.1
  • proof
  • Lemma 2: The tail of the series
  • proof
  • Lemma 3: The beginning of the series
  • proof
  • Definition 2
  • Lemma 4: The boundedness of the averaging operator
  • ...and 7 more