Some remarks on generalized Schwarz-Pick type inequality for harmonic quasiconformal mappings with simply connected ranges

The main result of this paper is a generalized Schwarz-Pick type inequality for ordinary harmonic quasiconformal mappings of the unit disk onto arbitrary simply connected domains in the complex plane. This result extends some of our earlier findings, as well as those presented in the excellent article [2]. Additionally, by analyzing the properties of the hyperbolic metric on simply connected hyperbolic domains in the complex plane, we establish the co-Lipschitz continuity of these mappings and determine the corresponding bi-Lipschitz constant with respect to the hyperbolic metric.