A result of holomorphic dependence for the eigenfunctions of a composition operator

In this paper we analyze the correspondence between a holomorphic self map $\varphi$ of the open complex unit disk $\mathbb{D}$ satisfying the conditions $\varphi (0)=0$, $0<|\varphi'(0)|<1$, $\mathrm{cl}\,\varphi (\mathbb{D})\subseteq\mathbb{D}$, and the corresponding principal eigenfunction $\sigma[\varphi]$ of the associated composition operator. We first prove that $\sigma[\cdot]$ is holomorphic in a suitable function space setting. Then we show that $\sigma[\cdot]$ is injective on the set of $\varphi$'s as above with a fixed value of $\varphi'(0)$ and we characterize the functions $\varphi$'s for which $\sigma[\cdot]$ has a local holomorphic inverse around $\varphi$.