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 Möbius transformation in Clifford analysis, the map** properties are given; The Jacobi determinant and the monogenic property under these Möbius transformations are shown. Finally, by using the above Möbius 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.
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Acknowledgements
Support by the National Natural Science Fund of China (No. 11471250) is gratefully acknowledged. The author is sincerely grateful to the reviewers for their helpful suggestions.
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Communicated by Uwe Kaehler.
The Project-sponsored by the NNSF of China (No. 11471250).
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Zhang, Z. The Schwarz Type Lemma in Upper Half Space in Clifford Analysis. Adv. Appl. Clifford Algebras 28, 98 (2018). https://doi.org/10.1007/s00006-018-0915-2
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DOI: https://doi.org/10.1007/s00006-018-0915-2