Landau-Toeplitz theorems for slice regular functions over quaternions (Q387723)

The authors give a generalization of the Schwarz lemma for slice regular functions and prove the corresponding Landau-Toeplitz theorem from 1907. These results base on former works by the authors, Della Rocchetta, D.C. Struppa as well as I. Sabadini and F. Colombo. The notion of a regular \(n\)-diameter is defined. Important Landau-Toeplitz inequalities can be shown for general \(n\geq 2\). The authors prove a complete generalization of the Landau-Toeplitz theorem for \(n=3\). A generalization to monogenic quaternions in Fueter's sense is still open.