On coefficient estimates for a class of holomorphic mappings

Let X be a complex Banach space with norm ‖ · ‖, B be the unit ball in X, D ⁿ be the unit polydisc in ℂ ⁿ . In this paper, we introduce a class of holomorphic mappings on B or D ⁿ . Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))⁻¹ f(x) ∈ and f(x) − x has a zero of order k + 1 at x = 0. We obtain coefficient estimates for f(x). These results unify and generalize many known results. This work was supported by National Natural Science Foundation of China (Grant No. 10571164), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20050358052), the Jiangxi Provincial Natural Science Foundation of China (Grant No. 2007GZS0177) and Specialized Research Fund for the Doctoral Program of Jiangxi Normal University.