THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES*

doi:10.1007/s10473-023-0509-5

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (5): 2075-2088.doi: 10.1007/s10473-023-0509-5

THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES^*

Qinghua XU^1,†, Weikang FANG¹, Weiheng FENG¹, Taishun LIU²

1. School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China;
2. Department of Mathematics, Huzhou University, Huzhou 313000, China

收稿日期:2022-02-24 修回日期:2023-04-20 发布日期:2023-10-25

THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES^*

Qinghua XU^1,†, Weikang FANG¹, Weiheng FENG¹, Taishun LIU²

1. School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China;
2. Department of Mathematics, Huzhou University, Huzhou 313000, China

Received:2022-02-24 Revised:2023-04-20 Published:2023-10-25
Contact: †Qinghua XU, E-mail: xuqh@mail.ustc.edu.cn
About author:Weikang FANG, E-mail:wkfang@163.com; Weiheng FENG, E-mail:whfeng@163.com; Taishun LIU, E-mail: lts@ustc.edu.cn
Supported by:
NNSF of China (11971165) and the Natural Science Foundation of Zhejiang Province (LY21A010003).

摘要/Abstract

摘要： Let $\mathcal{C}$ be the familiar class of normalized close-to-convex functions in the unit disk. In [17], Koepf demonstrated that, as to a function $f(\xi)=\xi+\sum\limits_{m=2}^\infty a_m\xi^m$ in the class $\mathcal{C}$, $$ \max\limits_{f\in \mathcal{C}}|a_3-\lambda a_2^2|\leq \left\{\begin{array}{ll} 3-4\lambda, \quad & \lambda\in[0, \frac{1}{3}], \\[3mm] \frac{1}{3}+\frac{4}{9\lambda}, \quad & \lambda\in[\frac{1}{3}, \frac{2}{3}], \\[3mm] 1, \quad & \lambda\in[\frac{2}{3}, 1]. \end{array}\right.$$ By applying this inequality, it can be proven that $||a_3|-|a_2||\leq 1$ for close-to-convex functions. Now we generalized the above conclusions to a subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.

关键词: Fekete and Szegö inequality, successive coefficients difference bound, close-to-starlike mappings, complex Banach space

Abstract: Let $\mathcal{C}$ be the familiar class of normalized close-to-convex functions in the unit disk. In [17], Koepf demonstrated that, as to a function $f(\xi)=\xi+\sum\limits_{m=2}^\infty a_m\xi^m$ in the class $\mathcal{C}$, $$ \max\limits_{f\in \mathcal{C}}|a_3-\lambda a_2^2|\leq \left\{\begin{array}{ll} 3-4\lambda, \quad & \lambda\in[0, \frac{1}{3}], \\[3mm] \frac{1}{3}+\frac{4}{9\lambda}, \quad & \lambda\in[\frac{1}{3}, \frac{2}{3}], \\[3mm] 1, \quad & \lambda\in[\frac{2}{3}, 1]. \end{array}\right.$$ By applying this inequality, it can be proven that $||a_3|-|a_2||\leq 1$ for close-to-convex functions. Now we generalized the above conclusions to a subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.

Key words: Fekete and Szegö inequality, successive coefficients difference bound, close-to-starlike mappings, complex Banach space

中图分类号:

32H30

Qinghua XU, Weikang FANG, Weiheng FENG, Taishun LIU. THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES^*[J]. 数学物理学报(英文版), 2023, 43(5): 2075-2088.

Qinghua XU, Weikang FANG, Weiheng FENG, Taishun LIU. THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2075-2088.

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THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES^*

THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES^*

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