Improved Bohr inequality for harmonic mappings

Abstract

Based on improving the classical Bohr inequality, we get in this paper some refined versions for a quasi-subordination family of functions, one of which is key to build our results. By means of these investigations, for a family of harmonic mappings defined in the unit disk $\D$, we establish an improved Bohr inequality with refined Bohr radius under particular conditions. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. % in a logical way. Here the family of harmonic mappings have 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 $\D$ and for some $k\in [0,1]$.

18 pages; The article is to appear in Mathematische Nachrichten