Let K be the familiar class of normalized convex functions in the unit disk. In [14], Keogh and Merkes proved that for a function $$f(z) = z + \sum\limits_{k = 2}^\infty {{a_k}} {z^k}$$ in the class K, $$\left| {{a_3} - \lambda a_2^2} \right| \le \max \left\{ {{1 \over 3},\left| {\lambda - 1} \right|} \right\},\;\;\;\;\lambda \in \mathbb{C}.$$ The above estimate is sharp for each λ. 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.

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关于若干复变量中拟凸映射子类的系数不等式的细化

令 K 为单位圆盘中熟悉的归一化凸函数类。在 [14] 中，Keogh 和 Merkes 证明了对于 K 类中的函数 $$f(z) = z + \sum\limits_{k = 2}^\​​infty {{a_k}} {z^k}$$ , $$\left| {{a_3} - \lambda a_2^2} \right| \le \max \left\{ {{1 \over 3},\left| {\lambda - 1} \right|} \right\},\;\;\;\;\lambda \in \mathbb{C}.$$ 上述估计对于每个 λ 都是尖锐的。在本文中，我们在 $$\mathbb{U}$$ 上建立归一化凸函数 f 的对应不等式，使得 z = 0 是 f(z) − z 的 k + 1 阶零，然后我们扩展这个结果到更高的维度。这些结果概括了一些已知的结果。