The 3-Point Spectral Pick Interpolation Problem and an Application to Holomorphic Correspondences

Abstract

We provide a necessary condition for the existence of a 3-point holomorphic interpolant F : D −→ Ωn, n ≥ 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 O(D, Ωn)-interpolation problem to establish a Schwarz lemma -which may be of independent interest -for holomorphic correspondences from D to a general planar domain Ω ⋐ C.

Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

“…This is shown in Observation 6.1 by explicitly computing the function B k, z (•). It turns out that when Ω = D the statement of Theorem 1.9 coincides with the statement of Theorem 1.4 in [4]. We also refer the reader to Remark 1.5 in [4] for the discussion of how Theorem 1.4 in [4] is different from the other results that are present in the literature related to the 3-point interpolation problem.…”

“…We shall present our proof of Theorem 1.9 in Section 6. The proof is strongly motivated from the proof of Theorem 1.4 in [4]. In fact, Theorem 1.9 above is a generalization of Theorem 1.4 in [4].…”

“…The proof is strongly motivated from the proof of Theorem 1.4 in [4]. In fact, Theorem 1.9 above is a generalization of Theorem 1.4 in [4]. This is shown in Observation 6.1 by explicitly computing the function B k, z (•).…”

“…We stated it here since it follows from the pluri-subharmonicity of spectral radius function; see e.g. [4,Section 4]. We are now ready to present our proof of Theorem 1.7.…”

“…The lemma is a consequence of the fact that ρ| Ωn is pluri-subharmonic. We do not wish to present a proof of the above lemma here; we refer the interested reader to [4,Lemma 4.3] for a proof. We also wish to state another another lemma which is a consequence of the pluri-subharmonicity of the spectral radius function.…”