Improved and Refined Bohr-Type Inequalities for Slice Regular Functions over Octonions

TL;DR

This paper extends Bohr inequalities to slice regular functions over octonions, providing generalized, improved, and refined versions with sharp bounds in a non-associative setting.

Contribution

It introduces the first generalization and refinement of Bohr inequalities for octonionic slice regular functions, advancing non-associative function theory.

Findings

Generalized Bohr inequality for octonionic slice regular functions

Improved versions of Bohr inequality with sharper bounds

Refined Bohr inequality for functions with real part bounded by 1

Abstract

A crucial extension of quaternionic function theory to octonions is the concept of slice regular functions, introduced to handle holomorphic-like properties in a non-associative setting. In this paper, first we present a generalization of the Bohr inequality, and improved versions of the Bohr inequality for slice regular functions over the largest alternative division algebras of octonions $\mathbb{O}$. Moreover, we provide a refined version of the Bohr inequality for slice regular functions $f$ on $\mathbb{B}$ such that $ {\rm Re}(f(x)) \leq 1 $ for all $x \in \mathbb{B}$. All the results are shown to be sharp.