Bohr phenomena for slice regular functions over Quaternions

TL;DR

This work extends Bohr's inequality to slice regular quaternionic functions, establishing sharp Bohr radii for the classes $S^*$, $C$, and $SB(B)$ and a generalized radius $R_m=\frac{m}{m+2}$ for $m\in(0,2]$. It uses coefficient bounds from quaternionic Schwarz lemmas and the regular-algebra toolkit to obtain extremal functions that prove sharpness, including radii $r_*=(3-\sqrt{5})/2$ and $1/2$ under suitable conditions. The authors further develop refined Bohr phenomena incorporating the quantity $S_q^* := \sum_{k\ge1} k|q^kp_k|^2$ and a polynomial-augmentation approach via $Q_N$, plus a low-radius bound $R_\ast\approx0.24683$ when the image lies in Re$(f)\le1$. These results broaden Bohr-type theory in noncommutative settings and provide sharp, generalizable inequalities for quaternionic slice regular spaces.

Abstract

Slice regular functions are a generalization of holomorphic functions to the setting of quaternions (and more generally, Clifford algebras). In this paper, we first establish the Bohr inequality for slice starlike functions and slice close-to-convex functions over quaternions $\mathbb{H}$. Next, we present a generalization of the Bohr inequality, and improved versions of the Bohr inequality for slice regular functions on the open unit ball $\mathbb{B}$ of $\mathbb{H}$. Finally, we provide a refined version of the Bohr inequality for slice regular functions $f$ on $\mathbb{B}$ such that $ {\rm Re}(f(q)) \leq 1 $ for all $q \in \mathbb{B}$. All the results are demonstrated to be sharp.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1
• Theorem 1.5
• Remark 1.2
• Remark 1.3
• Theorem 1.6
• Theorem 1.7