ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES

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ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES

ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES

ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES

ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES

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doi:10.1007/s10473-020-0603-x

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (6): 1653-1665.doi: 10.1007/s10473-020-0603-x

徐庆华, 赖元平

School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

收稿日期:2019-11-02 修回日期:2020-09-11 出版日期:2020-12-25 发布日期:2020-12-30
作者简介:Qinghua XU,E-mail:xuqh@mail.ustc.edu.cn;Yuanping LAI,E-mail:lyp@163.com
基金资助:
Supported by National Natural Science Foundation of China (11971165, 11561030).

Qinghua XU, Yuanping LAI

School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received:2019-11-02 Revised:2020-09-11 Online:2020-12-25 Published:2020-12-30
Supported by:
Supported by National Natural Science Foundation of China (11971165, 11561030).

摘要： Let $\mathcal{K}$ be the familiar class of normalized convex functions in the unit disk. In [14], Keogh and Merkes proved that for a function $f(z)=z+\sum\limits_{k=2}^\infty a_kz^k$ in the class $\mathcal{K}$ ,

\begin{aligned} | a_{3} - λ a_{2}^{2} | \leq max {\frac{1}{3}, | λ - 1 |}, λ \in C . \end{aligned}

$\begin{align*} |a_3-\lambda a_2^2|\leq \max \left\{\frac{1}{3}, |\lambda-1|\right\},\ \ \lambda \in \mathbb{C}. \end{align*}$ The above estimate is sharp for each

λ

$\lambda$ .
In this article, we establish the corresponding inequality for a normalized convex function

f

$f$ on

U

$\mathbb{U}$ such that

z = 0

$z=0$ is a zero of order

k + 1

$k+1$ of

f (z) - z

$f(z)-z$ , and then we extend this result to higher dimensions. These results generalize some known results.

关键词: Fekete-Szegö, problem, subclass of quasi-convex mappings, sharp coefficient bound

Abstract: Let $\mathcal{K}$ be the familiar class of normalized convex functions in the unit disk. In [14], Keogh and Merkes proved that for a function $f(z)=z+\sum\limits_{k=2}^\infty a_kz^k$ in the class $\mathcal{K}$ ,

\begin{aligned} | a_{3} - λ a_{2}^{2} | \leq max {\frac{1}{3}, | λ - 1 |}, λ \in C . \end{aligned}

$\begin{align*} |a_3-\lambda a_2^2|\leq \max \left\{\frac{1}{3}, |\lambda-1|\right\},\ \ \lambda \in \mathbb{C}. \end{align*}$ The above estimate is sharp for each

λ

$\lambda$ .
In this article, we establish the corresponding inequality for a normalized convex function

f

$f$ on

U

$\mathbb{U}$ such that

z = 0

$z=0$ is a zero of order

k + 1

$k+1$ of

f (z) - z

$f(z)-z$ , and then we extend this result to higher dimensions. These results generalize some known results.

Key words: Fekete-Szegö, problem, subclass of quasi-convex mappings, sharp coefficient bound

中图分类号:

32H30

徐庆华, 赖元平. ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES[J]. 数学物理学报(英文版), 2020, 40(6): 1653-1665.

Qinghua XU, Yuanping LAI. ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES[J]. Acta mathematica scientia,Series B, 2020, 40(6): 1653-1665.

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