Harmonic analogue of Bohr phenomenon of certain classes of univalent and analytic functions

Abstract

A class $ \mathcal {F} $ consisting of analytic functions $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ in the unit disk $ \mathbb {D}=\{z\in \mathbb {C}:\lvert z\rvert <1\} $ is said to satisfy Bohr phenomenon if there exists an $ r_f>0 $ such that $$ \sum _{n=1}^{\infty }\lvert a_n\rvert r^n\leq d(f(0),\partial \mathbb {D}) $$ for every function $ f\in \mathcal {F} $, and $\lvert z\rvert =r\leq r_f $. The largest radius $ r_f $ is known as the Bohr radius and the inequality $ \sum _{n=1}^{\infty }\lvert a_n\rvert r^n\leq d(f(0),\partial f(\mathbb {D})) $ is known as the Bohr inequality for the class $ \mathcal {F} $, where $d$ is the Euclidean distance. In this paper, we prove several sharp improved and refined versions of Bohr-type inequalities in terms of area measure of functions in a certain subclass of analytic and univalent (i.e. one-to-one) functions. As a consequence, we obtain several interesting corollaries on the Bohr-type inequality for the class which are the harmonic analogue of some Bohr-type inequality for the class of analytic functions.