On the elliptic harmonic mappings and sense-preserving harmonic mappings

In this paper, we first establish two versions of Landau-Bloch type theorem for (K, K') -elliptic harmonic mappings with a bounded minimum distortion. Next, we provide several coefficient estimates and a conjecture for (K, K') -elliptic harmonic mappings. Then, we establish three new versions of Landau-Bloch type theorem for sense-preserving harmonic mappings. Finally, we establish two sharp versions of Landau-Bloch type theorem for certain harmonic mappings. These results are sharp in some given cases and improve the related results of different authors.