Models of holomorphic functions on the symmetrized skew bidisc

The purpose of this paper is to develop the theory of holomorphic functions with modulus bounded by $1$ on the symmetrized skew bidisc $${\mathbb{G}}_r\{({\lambda }_1+r{\lambda }_2,r{\lambda }_1{\lambda }_2):{\lambda }_1\in \mathbb{D},{\lambda }_2\in \mathbb{D}\},$$ for a fixed $r\in (0,1)$. We show the existence of a realization formula and a model formula for such holomorphic functions.