On the elliptic harmonic mappings and sense-preserving harmonic mappings

In this paper, we first establish two versions of Landau–Bloch type theorem for \((K,K')\) $(K,K^{\prime })$ -elliptic harmonic mappings with a bounded minimum distortion. Next, we provide several coefficient estimates and a conjecture for \((K,K')\) $(K,K^{\prime })$ -elliptic harmonic mappings. Then, we establish three new versions of Landau–Bloch type theorem for sense-preserving harmonic mappings. Finally, we establish a sharp version 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.