Estimate on the Bloch constant for certain harmonic mappings under a differential operator

In this paper, we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the form L(f) = zf_z - z̅f_z̅ , where f represents normalized harmonic mappings with bounded dilation. Then, using these results, we present better estimations for the Bloch constants of certain harmonic mappings L ( f ), where f is a K -quasiregular harmonic or open harmonic. Finally, we establish three versions of Bloch-Landau type theorem for biharmonic mappings of the form L ( f ). These results are sharp in some given cases and improve the related results of earlier authors.