The inverse map satisfies $|\phi^{-1}(z)-\phi^{-1}(a)|\leq c|z-a|^{n/(n-2)}$
at any vertex $a$ (reciprocal to the exponent of the direct map). Since this the exponent is $>1$ the inverse is just Lipschitz
everywhere. You do not need any general theorems for this. At the vertex the inverse map behaves like a power. At all other points it is smooth (analytic).