Some Properties of the Slice Regular Schwarzians

The complex Schwarzian is a complex differential operator with several applications and properties in the theory of functions of one complex variable, such as the following: (a) Certain estimates on the complex Schwarzian allow us to find the local and the global univalence, and reciprocally, see Chuaqui et al. (Math Proc Camb Philos Soc 143, 2006. https://doi.org/10.1017/S0305004107000394), Nehari (Bull Am Math Soc 55:545–551, 1949), Osgood (Old and new on the Schwarzian derivative, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), Springer, New York, pp 275–308, 1998) and Schwarz (Trans Am Math Soc 80:159–186, 1955). (b) The growth of the complex Schwarzian shows a necessary condition to establish the quasiconformal extensions to the sphere of some holomorphic functions, see Osgood (1998). (c) The solutions of some differential equations defined from the complex Schwarzian are deeply related with the solutions of some Riccati equations, see Steinmetz (Ann Acad Sci Fenn 39:503–511, 2014). The aim of this work is to show that the slice regular Schwarzians defined in the theory of quaternionic slice regular functions, see González-Cervantes (Adv Appl Clifford Algebras 29:82, 2019. https://doi.org/10.1007/s00006-019-1004-x, satisfy properties similar to those commented above.