Linearly invariant families and harmonic Bloch mappings on hyperbolic domain

UDC 517.5 We extend the notions of harmonic Bloch, harmonic Bloch-type mappings, and a linearly invariant family of harmonic mappings defined in the open unit disc $\mathbb D$ of the complex plane to a hyperbolic domain of the complex plane. Then we establish the relation between linearly invariant family of holomorphic functions with harmonic Bloch mappings and a linearly invariant family of harmonic mappings with harmonic Bloch-type mappings defined on a hyperbolic domain of the complex plane. We also prove that the bounds of the Bloch seminorms for harmonic Bloch and harmonic Bloch-type mappings depend on the order of the linearly invariant family of holomorphic functions and the order of the linearly invariant family of harmonic mappings, respectively.