The Schwarz Type Lemma in Upper Half Space in Clifford Analysis

Abstract

In this paper, we firstly construct the Euler’s expression of hyper-complex numbers in $$H_{n+1}$$ , as an application of the Euler’s expression, we prove that the well-known homogeneous monogenic polynomials $$V_{l_1\dots l_p}(\mathbf{x})$$ are functions with values in $$H_{n+1}$$ . Secondly we construct a type of general Mobius transformation in Clifford analysis, the mapping properties are given; The Jacobi determinant and the monogenic property under these Mobius transformations are shown. Finally, by using the above Mobius transformation and modifying the Schwarz lemma in Zhang (J Math Anal Appl 443:1130–1141, 2016), we establish the Schwarz–Pick type lemmas for monogenic functions in the upper half space $$H^+_{n+1}$$ . A new generalization of the Schwarz lemma for harmonic functions with values in $$H_{n+1}$$ is also given.