$k$-monogenic functions and Möbius transformations

Clifford analysis is a fundamental framework for extending complex function theory to high-dimensional spaces, where $k$-monogenic functions (high-order generalizations of monogenic functions) play a pivotal role in geometric function theory and partial differential equations. However, there is a paucity of research findings regarding the Möbius transformations of $k$-monogenic functions for arbitrary $k$. In this paper, we first derived that the composite function constructed from a $k$-monogenic function and a Möbius transformation remains $k$-monogenic when $k=2,3,4$, and was generalized to the general case. Then, as applications, we proved the Schwarz-Pick-type lemma for harmonic functions using a new method, and we gave a version of the Schwarz-Pick-type lemma for inframonogenic functions. This work fills the gap in the transformation theory of $k$-monogenic functions, enriches the family of Schwarz-Pick-type lemmas in Clifford analysis, and provides theoretical tools to solve research related to high-dimensional geometric function theory.