The 3-point spectral Pick interpolation problem and an application to holomorphic correspondences. (English) Zbl 1436.30032
Summary: We provide a necessary condition for the existence of a 3-point holomorphic interpolant \(F:\mathbb{D}\longrightarrow \Omega_n\), \(n\ge 2\). 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 \(\mathcal{O}(\mathbb{D},\,\Omega_n)\)-interpolation problem to establish a Schwarz lemma – which may be of independent interest – for holomorphic correspondences from \(\mathbb{D}\) to a general planar domain \(\Omega \Subset\mathbb{C} \).
MSC:
| 30E05 | Moment problems and interpolation problems in the complex plane |
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