Linearly invariant families and harmonic Bloch mappings on hyperbolic domain - Bhowmik, Bappaditya; Sen, Sambhunath (Ukrains’kyi Matematychnyi Zhurnal, 2025)
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 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.