Nehari‐type families of harmonic mappings

We introduce affine and linearly invariant families of locally injective harmonic mappings of the unit disk D. We derive sharp distortion theorems for the Jacobian that are used to establish a uniform modulus of continuity for the quasiconformal mappings in each class. Finally, we find a converse of a recent theorem of Chen and Ponnusamy characterizing when the image f(D) under a quasiconformal harmonic univalent mapping is a John domain.