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

We introduce the notion of a \lambda-nonisotropically balanced domain and show that the symmetrized polydisc in C^n, n \geq 2, is an example of such a domain. Given a \lambda-nonisotropically balanced domain \Omega, we derive effective estimates from above and from below for the Lempert function at (0,z)\in\Omega\times\Omega. 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 \Omega_n, we deduce: a) a formula for the Lempert function at (0,W)\in\Omega_n\times\Omega_n; and b) a necessary and sufficient condition for realising a two-point Nevanlinna- Pick interpolation in the spectral unit ball.