The relationship between interior and exterior Riemann maps

Question

Let $U\subset\widehat{\mathbb{C}}$ be a compact Jordan domain with $\partial U$ connected. Is there a relationship between the exterior Riemann map associated to $U$, $$\varphi:\mathring{(\mathbb{D}^c)}\rightarrow U^c,\;\;\;\;\;\;\;\varphi(z)=az+f_0+f_1z^{-1}+...$$ and the interior Riemann map, $$\psi:\mathbb{D}\rightarrow\mathring{U},\;\;\;\;\;\;\;\psi(z)=g_0+g_1z+g_2z^2+...\;?$$ Is there, for example, a relationship between the coefficients of the Laurent series of $\varphi$ and $\psi$?