The article considers two properties of several metrics defined in the unit disk \(\Delta \) in \({\mathbb {C}}\) . One is the Lipschitz constant of Möbius transformation preserving \(\Delta \) , especially the t-metric case and the Hilbert metric case. In particular, we confirm a conjecture posed by Rainio and Vuorinen (Results Math 77:71–80, 2022) about t-metric, and disprove the conjecture posed by Rainio and Vuorinen (Stud Sci Math Hung 60:175–191, 2023) about Hilbert metric. Moreover, we show that the metric dimensions of metric spaces \((\Delta , d)\) , where d includes hyperbolic metric, j-metric, Hilbert metric and t-metric, are 3. Thus, we solve the open question posed by Bau and Beardon (Comput Methods Funct Theory 13:295–305, 2013) about these metrics in the unit disk case.
