I'm currently reading chapter 6 of Gilbarg + Trundinger book on second order elliptic pde. They define $C^{k,\alpha}$ domain in the following manner:
A domain $\Omega\subset\mathbb{R}^{n}$ and its boundary are said to be of class $C^{k,\alpha}$, $0< \alpha \leq 1$, if for each $x_0\in \partial\Omega$ there is a ball $B=B(x_0)$ and a one to one mapping $\psi$ of $B$ onto $D\subset \mathbb{R}^n$ such that: $$i) \,\, \psi(B\cap \Omega)\subset \mathbb{R}_{+}^{n};\,\,\,\,\,\,\,\,\,\, ii) \,\, \psi(B\cap \partial\Omega)\subset \partial\mathbb{R}_{+}^{n}; \,\,\,\,\,\,\,\,\,\, iii) \psi\in C^{k,\alpha}(B), \psi^{-1}\in C^{k,\alpha}(D).$$
Then they go on to casually claim that if $k\geq 1$ then $\Omega$ is locally the graph of $C^{k,\alpha}$ functions.
I dont see how to prove this statement, the typical $C^{k}$ version of the result can be proven using the inverse function theorem, but as far as I'm aware there is no $C^{k,\alpha}$ version of the inverse function theorem.
Can someone help indicate a path to prove this?