Distortion Theorems for Classes of g-Parametric Starlike Mappings of Real Order in ℂn

In this paper, we define the class S ^ g γ ( B X ) of g -parametric starlike mappings of real order γ on the unit ball B X in a complex Banach space X , where g is analytic and satisfies certain conditions. By establishing the distortion theorem of the Fréchet-derivative type of S ^ g γ ( B X ) with...

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Bibliographic Details
Published in:Acta mathematica scientia 2023-07, Vol.43 (4), p.1491-1502
Main Authors: Liu, Hongyan, Tu, Zhenhan, Xiong, Liangpeng
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
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Summary:In this paper, we define the class S ^ g γ ( B X ) of g -parametric starlike mappings of real order γ on the unit ball B X in a complex Banach space X , where g is analytic and satisfies certain conditions. By establishing the distortion theorem of the Fréchet-derivative type of S ^ g γ ( B X ) with a weak restrictive condition, we further obtain the distortion results of the Jacobi-determinant type and the Fréchet-derivative type for the corresponding classes (compared with S ^ g γ ( B X ) ) defined on the unit polydisc (resp. unit ball with the arbitrary norm) in the space of n -dimensional complex variables, n ⩾ 2. Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space. The main theorems also generalize and improve some recent works.
ISSN:0252-9602
1572-9087
DOI:10.1007/s10473-023-0402-2