Landau-Bloch type theorem for elliptic and $K$-quasiregular harmonic mappings

In this paper, we establish an improved coefficient bounds for quasiregular and elliptic harmonic mappings and using these bounds we establish Landau-Bloch type theorem for $(K,K^{\mathrm{\prime }})$-elliptic and K-quasiregular harmonic mappings in plane. Furthermore, we prove the coefficient estimates for $K$-quasiconformal harmonic self maps defined on the unit disk $\mathbb{D}$.