Julia theory for slice regular functions (Q2833001)

The extension of a lemma of Schwarz by \textit{G. Julia} [Acta Math. 42, 349--355 (1919; JFM 47.0272.01)] and its extension by \textit{C. Carathéodory} [Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1929, 39--54 (1929; JFM 55.0209.02)] have been fundamental to the study of iteration of holomorphic self-mappings of the open unit disc in the complex plane. Here the authors consider slice regular self-mappings of the open unit ball \(B\) in the quaternions obtaining a corresponding Julia lemma and Julia-Caratheodory theorem. Starting with the counterpart of the Schwarz-Pick theorem in the quaternions setting as established by \textit{C. Bisi} and \textit{C. Stoppato} [Indiana Univ. Math. J. 61, No. 1, 297--317 (2012; Zbl 1271.30024)], the authors also establish the Burns-Krantz rigidity theorem and the boundary Schwarz lemma for slice regular self-mappings of \(B\). With the exception of the rigidity results, the proofs do not proceed by tracing the form of the complex setting. Further an example shows that the slice derivative of a slice regular self-mapping of \(B\) at a boundary fixed point need not be positive (in contrary to what might have been expected from the complex setting). A quaternionic version of Lindelöf's principle which follows from the corresponding complex setting is used to establish the Julia-Caratheodory theorem which in turn is the basis for the proof of Hopf's lemma (an improved version of the Schwarz theorem when the boundary point one in \(\partial B\) is considered). An overview of boundary behavior works is provided in the introduction of this well-written manuscript. The inclusion in the paper of some basics concerning slice regular functions adds to the readability of the paper.