Abstract
In this paper, first, a sufficient condition for almost convex mappings of order \(\alpha \) defined on the unit ball of complex Hilbert spaces and another sufficient condition for almost quasi-convex mappings of order \(\alpha \) defined on the unit ball of complex Banach spaces are given. Second, the distortion theorem of the Fréchet derivative for almost convex mappings of order \(\alpha \) on the unit ball of complex Banach spaces, the homogeneous ball of complex Banach spaces, and the unit ball of complex Hilbert spaces are established respectively. Finally, the distortion theorem of the Jacobi determinant for almost convex mappings of order \(\alpha \) on the Euclidean unit ball in \(\mathbb {C}^n\) is obtained. Our results generalize many known results.
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Acknowledgments
This work was supported by the Key Program of National Natural Science Foundation of China (Grant No.11031008), the National Natural Science Foundation of China (Grant Nos.10971063, 11061015, 11471111), and Guangdong Natural Science Foundation (Grant No.2014A030307016). The authors thank the referee for helpful comments and suggestions.
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Communicated by Saminathan Ponusammy.
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Liu, X., Liu, T. & Xu, Q. Distortion Theorems for Almost Convex Mappings of Order \(\alpha \) in Several Complex Variables. Bull. Malays. Math. Sci. Soc. 39, 1363–1377 (2016). https://doi.org/10.1007/s40840-015-0224-z
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DOI: https://doi.org/10.1007/s40840-015-0224-z
Keywords
- Distortion theorem
- Fréchet derivative
- Jacobi determinant
- Almost convex mapping of order \(\alpha \)
- Almost quasi-convex mapping of order \(\alpha \)