A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings

Abstract

In this paper, we prove a Schwarz lemma at the boundary for holomorphic mappings f between Hilbert balls, and obtain related consequences. Especially, we obtain estimations of ∥Df(z0)∥ on the holomorphic tangent space for holomorphic mappings f or for homogeneous polynomial mappings f between Hilbert balls. Next, we prove the boundary rigidity theorem for holomorphic self-mappings of a Hilbert ball which have an interior fixed point. We obtain two distortion theorems for various subclasses of starlike mappings on the Euclidean unit ball $${\mathbb{B}^n}$$ in ℂn, as applications of the boundary Schwarz lemma for holomorphic mappings between the Euclidean unit balls. Finally, certain coefficient bounds for subclasses of starlike mappings on the unit ball of a complex Hilbert space are also obtained as an application of the estimation of ∥Df(z0)∥ on the holomorphic tangent space for homogeneous polynomial mappings f between Hilbert balls.