arXiv:1209.2060v1 [math.CV] 10 Sep 2012 The Schwarz-Pick lemma for slice regular functions Cinzia Bisi * Universit` a degli Studi di Ferrara Dipartimento di Matematica e Informatica Via Machiavelli 35, 44121 Ferrara, Italy cinzia.bisi@unife.it Caterina Stoppato *† Universit` a degli Studi di Milano Dipartimento di Matematica “F. Enriques” Via Saldini 50, 20133 Milano, Italy caterina.stoppato@unimi.it 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. 1 Introduction In the complex case, holomorphy plays a crucial role in the study of the intrinsic geometry of the unit disc Δ = {z ∈ C : |z | < 1} thanks to the Schwarz-Pick lemma [17, 18]. Theorem 1.1. Let f :Δ → Δ be a holomorphic function and let z 0 ∈ Δ. Then      f (z ) − f (z 0 ) 1 − f (z 0 )f (z )      ≤     z − z 0 1 − ¯ z 0 z     , (1) * Partially supported by GNSAGA of the INdAM and by FIRB “Geometria Differenziale Complessa e Dinamica Olomorfa”. † Partially supported by FSE and by Regione Lombardia. 1