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

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.