  In this paper, we construct the Funk-Finsler structure in various models of
the hyperbolic plane. In particular, in the unit disc of the Klein model, it
turns out to be a Randers metric, which is a non-Berwald Douglas metric.
Further, using Finsler isometries we obtain the Funk-Finsler structures in
other models of the hyperbolic plane. Finally, we also investigate the geometry
of this Funk-Finsler metric by explicitly computing the S-curvature, Riemann
curvature, flag curvature, and Ricci curvature in the Klein unit disc.
