Improved Bohr inequality for harmonic mappings

In order to improve the classical Bohr inequality, we explain some refined versions for a quasi-subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk $\mathbb{D}$. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form $f=h+\overline{g}$, where $g(0)=0$, the analytic part h is bounded by 1 and that $|g^{\prime }(z)|\le k|h^{\prime }(z)|$ in $\mathbb{D}$ and for some $k\in [0,1]$.