Improved Bohr's inequality for mappings defined on simply connected domains

Abstract

In this paper, we study the Bohr phenomenon for functions that are defined on a general simply connected domain of the complex plane. We improve known results of R. Fournier and St. Ruscheweyh for a class of analytic functions. Furthermore, we examine the case where a harmonic mapping is defined in a disk containing $\mathbb{D}$ and obtain a Bohr type inequality.