Bohr phenomenon for analytic and harmonic mappings on shifted disks

TL;DR

This paper establishes sharp bounds for Bohr inequalities and their refinements for analytic and harmonic functions defined on shifted disks, extending classical results to new geometric settings.

Contribution

It introduces new sharp Bohr-type inequalities for functions on shifted disks, broadening the scope of Bohr phenomenon studies.

Findings

Sharp Bohr inequalities for analytic functions on shifted disks

Refined and improved Bohr inequalities established

Results extend classical Bohr bounds to new geometric domains

Abstract

The primary objective of this paper is to establish several sharp results concerning the Bohr inequality, the refined Bohr inequality, and the improved Bohr inequality for the classes of analytic functions and harmonic mappings defined on the shifted disks \[ \Omega_{\gamma}=\left\{z\in\mathbb{C}:\left|z+\frac{\gamma}{1-\gamma}\right|<\frac{1}{1-\gamma}\right\}\quad\text{for}\quad\gamma\in[0,1).\]