Bohr's theorem for Cesáro operator and certain integral transforms over octonions

In this paper, we first establish the Bohr’s theorem for Cesáro 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.

💡 Research Summary

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The manuscript investigates Bohr‑type phenomena for a variety of operators acting on slice‑regular functions defined on the open unit ball ( \mathbb B\subset\mathbb O), where ( \mathbb O) denotes the octonion algebra. The class ( \mathcal{SRB}(\mathbb B)) consists of slice‑regular functions (f(x)=\sum_{k=0}^{\infty}x^{k}a_{k}) with the supremum norm (|f(x)|\le1) throughout ( \mathbb B). Building on Xu’s recent extension of the classical Bohr theorem to octonions (Theorem B), the authors establish sharp Bohr‑type inequalities for several linear operators: a β‑Cesàro operator, a Bernardi operator, the Libera operator, the Alexander operator, and two discrete integral transforms (Fourier and Laplace).

Main contributions

1. β‑Cesàro operator (T^{*}_{\beta}