Abstract
In this paper, we consider linear combinations of harmonic K-quasiregular mappings $f_j=h_j+\overline{g}_j$ $(j=1, 2)$ of the class ${\mathrm{Har}} (k; \phi_j)$, where $k\in [0,1)$, $\|\omega_{f_j}\|_{\infty}=\|g'_j/h'_j\|_{\infty}\leq k<1$, $k=(1-K)/(1+K)$, and $\phi_j=h_j+e^{i\theta}g_j$ is a univalent analytic function. We provide sufficient conditions for the linear combinations of mappings in these classes to be univalent and for the image domains to be linearly connected. Furthermore, we consider under which conditions the linear combination $f$ is bi-Lipschitz.
Funding Statement
The first and second authors were supported by NSF of China (Grant No. 11971124) and NSF of Guangdong Province (Grant No. 2021A1515010326). The third author was supported by NSF of China (Grant No. 12271189, 11971182), NSF of Fujian Province (Grant No. 2021J01304, 2019J01066) and Fujian Alliance of Mathematics (Grant No. 2023SXLMMS07).
Acknowledgment
We would like to thank the anonymous referee for his/her helpful comments that substantially improved the paper.
Citation
Jie Huang. Antti Rasila. Jian-Feng Zhu. "Injectivity criteria of linear combinations of harmonic quasiregular mappings." Kodai Math. J. 47 (1) 52 - 66, March 2024. https://doi.org/10.2996/kmj47104
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