Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quaternionic Nevanlinna Functions

Muhammad Ammar

Abstract

Nevanlinna theory studies the value distribution of meromorphic functions and provides powerful results in the form of the First and Second Main Theorems. In this paper, we introduce quaternionic analogues of the Nevanlinna functions. Starting from the Jensen formula due to Perotti (arXiv:1902.06485), we derive a notion of total order and an associated integrated counting function. We further define quaternionic Weil functions and corresponding mean proximity functions. In this context, we introduce the class of mean proximity balanced functions, which includes the slice-preserving functions and all semiregular functions with a dominating index in their power series. To address the failure of $\log|f^s|$ to be harmonic, we define a Harmonic Remainder Function that compensates for this defect in the Jensen formula. We then prove a weak First Main Theorem--type result for general semiregular functions and obtain a full First Main Theorem for the mean proximity balanced functions.

Quaternionic Nevanlinna Functions

Abstract

Nevanlinna theory studies the value distribution of meromorphic functions and provides powerful results in the form of the First and Second Main Theorems. In this paper, we introduce quaternionic analogues of the Nevanlinna functions. Starting from the Jensen formula due to Perotti (arXiv:1902.06485), we derive a notion of total order and an associated integrated counting function. We further define quaternionic Weil functions and corresponding mean proximity functions. In this context, we introduce the class of mean proximity balanced functions, which includes the slice-preserving functions and all semiregular functions with a dominating index in their power series. To address the failure of to be harmonic, we define a Harmonic Remainder Function that compensates for this defect in the Jensen formula. We then prove a weak First Main Theorem--type result for general semiregular functions and obtain a full First Main Theorem for the mean proximity balanced functions.
Paper Structure (14 sections, 21 theorems, 128 equations, 1 figure, 1 table)

This paper contains 14 sections, 21 theorems, 128 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The Nevanlinna Functions and Theorems
  3. Slice Regularity
  4. Lemmas on Slice Regularity
  5. A Quaternionic Jensen Formula
  6. Total Order
  7. Nevanlinna Functions
  8. Integrated Counting Function
  9. Mean Proximity Functions
  10. Harmonic Remainder Function
  11. Characteristic Function
  12. A First Main Theorem
  13. Open Questions
  14. Numerical Verification of the Jensen Formula

Key Result

Theorem 2.4

Let $a\in{\mathbb{P}^1(\mathbb{C})}$ and let $f\not\equiv a,\infty$ be a meromorphic function on $\mathbf{D}(R)$, $R\leq\infty$. Then for all $r\leq R$, where $T(f,r)\coloneq T(f,\infty,r)$.

Figures (1)

  • Figure 1: The complex line (slice) $L_I$. When restricted to the imaginary axes, $L_I$ is simply a line passing through the imaginary unit $I$ on ${\mathbb{S}}$. The line functions as the imaginary axis of the full slice.

Theorems & Definitions (83)

  • Definition 2.1: Integrated Counting Function
  • Definition 2.2: Mean Proximity Function
  • Definition 2.3: Nevanlinna Characteristic Function
  • Theorem 2.4: First Main Theorem
  • Corollary 2.5
  • Theorem 2.6: Second Main Theorem
  • Definition 3.1
  • Definition 3.2
  • Example 3.3: GENTILI2007279, Example 1.11
  • Definition 3.4: GENTILI2007279, Definition 1.12
  • ...and 73 more