Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings f=h+g¯ of the unit disk D, where h and g are analytic with g(0)=0, and determine the Bohr radius if any one of the following conditions holds: 1.h is bounded in D. 2.h satisfies the condition Re h(z)≤1 in D with h(0)>0. 3.both h and g are bounded in D. 4.h is bounded and g′(0)=0. We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in D is strictly less than 1. In addition, we determine the Bohr radius for the space B of analytic Bloch functions and the space BH of harmonic Bloch functions. The paper concludes with two conjectures.