Improved Bohr's inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr's inequality for the harmonic mappings of the form [Formula presented], where h is bounded by 1 and |g′(z)|≤|h′(z)|. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk [Formula presented] under the mapping f is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.