The main aim of this article is to establish analogues of Landau’s theorem for solutions to the $\overline{\unicode[STIX]{x2202}}$-equation in Dirichlet-type spaces.

When $G$ is a finite solvable group, we prove that $|G|$ can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$ is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.

Our main aim is to investigate the properties of harmonic ν-Bloch mappings. Firstly, we establish coefficient estimates and a Landau theorem for harmonic ν-Bloch mappings, which are generalizations of the corresponding results in Bonk et al. [‘Distortion theorems for Bloch functions’, Pacific. J. Math.179 (1997), 241–262] and Chen et al. [‘Bloch constants for planar harmonic mappings’, Proc. Amer. Math. Soc.128 (2000), 3231–3240]. Secondly, we obtain an improved Landau theorem for bounded harmonic mappings. Finally, we obtain a Marden constant for harmonic mappings.