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 \(f=h+\overline{g}\), where \(h\) is bounded by 1 and \(|g'(z)|\le|h'(z)|\). The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. \cite{KayPon2}, for example a term related to the area of the image of the disk \(D(0,r)\) 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.