In this paper, we first give a coefficient inequality for holomorphic functions on the unit disc \({\mathbb {U}}\) in \({\mathbb {C}}\) which are subordinate to a holomorphic function p on \({\mathbb {U}}\) with \(p'(0)\ne 0\). Next, as applications of this theorem, we will give the Fekete-Szegö inequality for subclasses of normalized starlike mappings and normalized quasi-convex mappings of type B on the unit ball \({\mathbb {B}}\) of a complex Banach space. We also give the Fekete-Szegö inequality for \((1+r)J_r\), where \(J_r=J_r[f]\) is the nonlinear resolvent of a mapping f in the Carathéodory family \({{\mathscr {M}}}({\mathbb {B}})\). Various particular cases will be also considered.

在本文中，我们首先给出对于全纯函数的系数不等式单位圆上\（{\ mathbb【U}} \）在\（{\ mathbb {C}} \） ，其是从属于一个全纯函数p上\ ({\mathbb {U}}\)与\(p'(0)\ne 0\)。接着，作为该定理的应用中，我们将给出归一化的星形映射的子类的Fekete-Szegö不等式和类型的归一化拟凸映射乙单位球上的\（{\ mathbb {B}} \）一个复Banach的空间。我们还给出了\((1+r)J_r\)的 Fekete-Szegö 不等式，其中\(J_r=J_r[f]\)是映射f的非线性求解在 Carathéodory 家族中\({{\mathscr {M}}}({\mathbb {B}})\)。还将考虑各种特殊情况。