Let E be the unit ball of a complex Banach space, \(\mathcal {M}_g(E)\) be a class of holomorphic mappings on E (see Definition 1.2). Let F be a holomorphic mapping on E, and have the power series expansion \(F(x)=x+\sum \limits _{l=k+1}^{\infty }\frac{D^{l}F(0)(x^{l})}{l!}\) for x near the origin, where k is a positive integer. In this paper, we establish various Fekete and Szegö inequalities for F such that F is locally biholomorphic on E and \((DF(x))^{-1}(D^2F(x)(x^2)+DF(x)(x))\in \mathcal {M}_g(E)\) . The results presented here generalize some known results in [10, 21] and [24].
