The Schwarz-Pick lemma for slice regular functions

Abstract

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.

to appear in Indiana Univ. Math. J., 21 pages

References in corpus (5)

• Slice regular functions on real alternative algebras
• Power series and analyticity over the quaternions
• A new series expansion for slice regular functions
• Regular Moebius transformations of the space of quaternions
• Poles of regular quaternionic functions

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• Slice-quaternionic Hopf surfaces
• The harmonicity of slice regular functions
• A Landau's theorem in several complex variables
• Landau's theorem for slice regular functions on the quaternionic unit ball
• Slice starlike functions over quaternions
• Some notions of subharmonicity over the quaternions
• On two Bloch type theorems for quaternionic slice regular functions
• Multipoint Schwarz-Pick Lemma for the quaternionic case
• Riemann slice-domains over quaternions I
• Boundary Julia theory for slice regular functions
• On the Quadratic Cone of ${\mathbb{R}}_3.$
• On geometric aspects of quaternionic and octonionic slice regular functions
• Julia theory for slice regular functions