The Schwarz lemma for functions with values in C(Vn,0)

In this paper, we first construct a type of Möbius transformations with Clifford coefficients and give some of its properties, for example: the mapping properties, preserving the symmetric points with respect to the sphere, the Jacobi determinant, the monogenic properties under these Möbius transformations. Then by using the integral representation formulas for harmonic functions with values in a Clifford algebra C(Vn,0)C(Vn,0) and the integral inequality estimation, we give the Schwarz lemma for harmonic functions with values in a Clifford algebra C(Vn,0)C(Vn,0), which is sharper than the known results. Combining the Schwarz lemma with the Möbius transformations, the Schwarz–Pick type lemmas for functions with values in C(Vn,0)C(Vn,0) are given. These results greatly improve the results in [21].