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

doi:10.1007/s10473-020-0603-x

Abstract

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 $\mathcal{K}$ ,

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

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

$f$ on

$\mathbb{U}$ such that

$z=0$ is a zero of order

$k+1$ of

$f(z)-z$ , 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

CLC Number:

32H30

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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