Bohr radius for locally univalent harmonic mappings

We consider the class of all sense-preserving harmonic mappings $f= h+\overline{g}$oftheunitdisk$\ID$,where$h$and$g$areanalyticwith$g(0)=0$, and determine the Bohr radius if any one of the following conditions holds: \bee $h$ is bounded in $\text{ID}$. $h$ satisfies the condition ${\rm Re}, h(z)\leq 1$in$$\mathbb{D}$$with$h(0)>0$\mathrm{.}both$h$and$g$areboundedin$\ID$\mathrm{.}$h$isboundedand$g’(0)=0$. \eee We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of $f$ in $\text{ID}$ is strictly less than $1$. In addition, we determine the Bohr radius for the space $\mathcal{B}$ of analytic Bloch functions and the space ${\mathcal{B}}_H$ of harmonic Bloch functions. The paper concludes with two conjectures.