Bohr's theorem for Ces\'aro operator and certain integral transforms over octonions

TL;DR

This paper extends Bohr's theorem to slice regular functions over octonions, establishing inequalities for various operators and integral transforms, with all results proven to be sharp.

Contribution

It introduces Bohr's theorem for Cesàro and other operators on octonionic slice regular functions, a novel extension in non-associative algebra settings.

Findings

Bohr's theorem established for Cesàro operator on octonionic slice regular functions.

Bohr type inequalities derived for Bernardi, Libera, and Alexander operators.

Sharp Bohr-type inequalities obtained for Fourier and Laplace transforms.

Abstract

In this paper, we first establish the Bohr's theorem for Ces\'aro operator defined for $f\in \mathcal{SRB}(\mathbb{B})$ of slice regular functions in the open unit ball $\mathbb{B}$ of the largest alternative division algebras of octonions $\mathbb{O}$, such that $|f(x)| \leq 1$ for all $x \in \mathbb{B}$. Next, we establish Bohr type inequalities for Bernardi operator for the functions $f\in \mathcal{SRB}(\mathbb{B})$, and with the help of this, we obtain Bohr type inequality for Libera operator and Alexander operator. Finally, we obtain Bohr-type inequalities for certain integral transforms, namely Fourier (discrete) and Laplace (discrete) transforms for $f\in \mathcal{SRB}(\mathbb{B}).$ All the results are proven to be sharp.