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

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 $\mathrm{R}\mathrm{e}(f(x))\le 1$ for all $x\in \mathbb{B}$. All the results are shown to be sharp.