Nehari-type families of harmonic mappings

We introduce affine and linearly invariant families of locally injective harmonic mappings of the unit disk . 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 under a quasiconformal harmonic univalent mapping is a John domain.