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.
Bhowmik et al. (Wed,) studied this question.
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