Bohr phenomenon for certain classes of harmonic mappings

Abstract

Bohr phenomenon for analytic functions f, where f(z)=∑n=0∞anzn, was first introduced by Harald Bohr in 1914 and deals with finding the largest radius rf, 0<rf<1, such that the inequality ∑n=0∞|an||z|n<1 holds for |z|=r≤rf whenever |f(z)|<1 holds in the unit disk 𝔻={z∈ℂ:|z|<1}. The Bohr phenomenon for harmonic functions of the form f(z)=h+ḡ, 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, where d(f(0),∂f(𝔻)) is the Euclidean distance between f(0) and the boundary of f(𝔻). We prove several improved versions of the sharp Bohr radius for the classes of harmonic and univalent functions. Further, we prove several corollaries as a consequence of the main results.