Schwarz Lemmas via the Pluricomplex Green’s Function

Abstract

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green's function. We also prove several other Schwarz lemmas using this function.In this section, we de ne and prove basic facts about an extremal function de ned using plurisubharmonic functions. Our treatment is from [Kob98, p. 184] where the de nition is attributed to Klimek [Kli85]. The paper by Demailly [Dem87] contains further properties of this function.De nition 4. Let X be a complex manifold. Fix z 0 ∈ X and de ne the extremal functionwhere P X (z 0 ) is the collection of functions ϕ on X that satisfy:1. ϕ is upper semi-continuous,

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

“…For n ∈ Z + , recall the spectral unit ball, Ω n ⊂ C n 2 , is the collection of all matrices A ∈ M n (C) whose spectrum σ(A) is contained in D. We also recall the definition of spectral radius ρ of a matrix A defined by ρ(A) := max | λ | : λ ∈ σ(A) . We first state the result: Result 4.1 (Janardhanan, [11]). The spectral unit ball, Ω n , is an unbounded, balanced, pseudo-convex domain with Minkowski function given by the spectral radius ρ.…”