The Schwarz lemma in Clifford analysis

Abstract

In this paper, we first give the Cauchy type integral representation for harmonic functions in the Clifford analysis framework, and by using integral representations for harmonic functions in Clifford analysis, the Poisson integral formula for harmonic functions is represented. As its application, the mean value theorems and the maximum modulus theorem for Clifford-valued harmonic functions are presented. Second, some properties of Möbius transformations are given, and a close relation between the monogenic functions and Möbius transformations is shown. Finally, by using the integral representations for harmonic functions and the properties of Möbius transformations, Schwarz type lemmas for harmonic functions and monogenic functions are established.