The 3-point spectral Pick interpolation problem and an application to holomorphic correspondences

We provide a necessary condition for the existence of a 3-point holomorphic interpolant $F:\mathbb{D}⟶{\mathrm{\Omega }}_n$, $n\ge 2$. Our condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma. We combine some of the ideas and techniques used in our result on the $\mathcal{O}(\mathbb{D},{\mathrm{\Omega }}_n)$-interpolation problem to establish a Schwarz lemma – which may be of independent interest – for holomorphic correspondences from $\mathbb{D}$ to a general planar domain $\Omega\Subset $\mathbb{C}$$.