Bohr phenomenon for certain subclasses of harmonic mappings

Abstract

The Bohr phenomenon for analytic functions of the form f(z)=∑n=0∞anzn, first introduced by Harald Bohr in 1914, deals with finding the largest radius rf, 0<rf<1, such that the inequality ∑n=0∞|anzn|≤1 holds whenever the inequality |f(z)|≤1 holds in the unit disk D={z∈C:|z|<1}. The exact value of this largest radius known as Bohr radius, which has been established to be rf=1/3. The Bohr phenomenon [1] for harmonic functions f of the form f(z)=h(z)+g(z)‾, where h(z)=∑n=0∞anzn and g(z)=∑n=1∞bnzn is to find the largest radius rf, 0<rf<1 such that∑n=1∞(|an|+|bn|)|z|n≤d(f(0),∂f(D)) holds for |z|≤rf, here d(f(0),∂f(D)) denotes the Euclidean distance between f(0) and the boundary of f(D). In this paper, we investigate the Bohr radius for several classes of harmonic functions in the unit disk D.