Let $\mathcal{C}$ be the familiar class of normalized close-to-convex functions in the unit disk. In Koepf [On the Fekete-Szegö problem for close-to-convex functions. Proc Amer Math Soc. 1987;101:89–95], Koepf proved that for a function $f(z)=z+\sum _{k=2}^{\mathrm{\infty }}{a}_k{z}^k$ in the class $\mathcal{C}$, $|a_3-\lambda a_2^2|\le \{\begin{array}{ll}3-4\lambda ,& \lambda \in [0,\frac{1}{3}],\\ \frac{1}{3}+\frac{4}{9\lambda },& \lambda \in [\frac{1}{3},\frac{2}{3}],\\ 1,& \lambda \in [\frac{2}{3},1].\end{array}$ As an important application, in the same paper, Koepf showed that $||a_3|-|a_2||\le 1$ for close-to-convex functions. In this paper, we extend the above results to a subclass of close-to-quasi-convex mappings of type B defined on the unit polydisc in ${\mathbb{C}}^n$, and establish the sharp difference bound for the second and third coefficients of homogeneous expansions for this class of holomorphic mappings. The results presented here would provide a new path for solving the Bieberbach conjectures in several complex variables.

让$\mathcal{C}$是单位圆盘中熟悉的归一化近凸函数类。在 Koepf [关于接近凸函数的 Fekete-Szegö 问题。Proc Amer 数学 Soc。1987;101:89–95], Koepf 证明了对于一个函数$F(z)=z+\sum _{k=\mathrm{2个}}^{\mathrm{\infty }}{\mathrm{一种}}_k{z}^k$在课堂里$\mathcal{C}$,$|{\mathrm{一种}}_{\mathrm{3个}}-\lambda {\mathrm{一种}}_{\mathrm{2个}}^{\mathrm{2个}}|\le \{\begin{array}{ll}\mathrm{3个}-\mathrm{4个}\lambda ,& \lambda \in [0,\frac{\mathrm{1个}}{\mathrm{3个}}],\\ \frac{\mathrm{1个}}{\mathrm{3个}}+\frac{\mathrm{4个}}{9\lambda },& \lambda \in [\frac{\mathrm{1个}}{\mathrm{3个}},\frac{\mathrm{2个}}{\mathrm{3个}}],\\ \mathrm{1个},& \lambda \in [\frac{\mathrm{2个}}{\mathrm{3个}},\mathrm{1个}].\end{array}$作为一个重要的应用，在同一篇论文中，Koepf 表明$||{\mathrm{一种}}_{\mathrm{3个}}|-|{\mathrm{一种}}_{\mathrm{2个}}||\le \mathrm{1个}$对于接近凸函数。在本文中，我们将上述结果扩展到在单位多圆盘上定义的类型B的近准凸映射的子类${\mathbb{C}}^n$，并为此类全纯映射的齐次展开的第二和第三系数建立尖差界。这里给出的结果将为解决多个复变量中的 Bieberbach 猜想提供一条新途径。