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) - (z) over barf((z) over bar), 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 BlochLandau 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.