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.
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
.
DOI: 10.1007/s10473-020-0603-x
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