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 $$\mathbb{G}_{r} \stackrel{\rm def}{=} \Big\{( \lambda_{1}+r\lambda_{2} ,r\lambda_{1}\lambda_{2}): \lambda_{1}\in \mathbb{D}, \lambda_{2}\in\mathbb{D}\Big\},$$ for a fixed $r \in (0,1)$. We show the existence of a realization formula and a model formula for such holomorphic functions.