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Nonisotropically balanced domains, Lempert function estimates, and the spectral Nevanlinna-Pick problem

Gautam Bharali
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Abstract

We introduce the notion of a λ-nonisotropically balanced domain and show that the symmetrized polydisc in C^n, n \geq 2, is an example of such a domain. Given a λ-nonisotropically balanced domain Ω, we derive effective estimates from above and from below for the Lempert function at (0,z)\inΩ\timesΩ. We use these estimates to derive certain conditions for realising a two-point Nevanlinna-Pick interpolation in the symmetrized polydisc. Applying the ideas used in the derivation of our Lempert function estimates to the so-called spectral unit ball Ω_n, we deduce: a) a formula for the Lempert function at (0,W)\inΩ_n\timesΩ_n; and b) a necessary and sufficient condition for realising a two-point Nevanlinna- Pick interpolation in the spectral unit ball.

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Gautam Bharali. 2006-03-30. Nonisotropically balanced domains, Lempert function estimates, and the spectral Nevanlinna-Pick problem. https://arxiv.org/abs/math/0601107

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