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

TL;DR

This paper introduces a new necessary condition for 3-point holomorphic interpolation in complex domains, involving a simplified inequality, and applies it to develop a Schwarz lemma for holomorphic correspondences.

Contribution

It provides a novel necessary condition for 3-point interpolation that differs from existing conditions and extends Schwarz lemma concepts to holomorphic correspondences.

Findings

New necessary inequality condition for 3-point interpolation

A Schwarz lemma for holomorphic correspondences established

Application to holomorphic correspondences from the disk to planar domains

Abstract

We provide a necessary condition for the existence of a 3-point holomorphic interpolant $F:\mathbb{D}\longrightarrow\Omega_n$, $n\geq 2$. 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 $\mathcal{O}(\mathbb{D},\Omega_n)$-interpolation problem to establish a Schwarz lemma -- which may be of independent interest -- for holomorphic correspondences from $\mathbb{D}$ to a general planar domain $\Omega\Subset \mathbb{C}$.