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.
Citation: Xiaotong Liang, Chunxue Duan, Zihan Su, Yonghong Xie. $ k $-monogenic functions and Möbius transformations[J]. AIMS Mathematics, 2025, 10(12): 29132-29150. doi: 10.3934/math.20251281
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.
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Xiaotong Liang, Chunxue Duan, Zihan Su, Yonghong Xie. $ k $-monogenic functions and Möbius transformations[J]. AIMS Mathematics, 2025, 10(12): 29132-29150. doi: 10.3934/math.20251281
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