On the coefficients of concave univalent functions

摘要

Let D denote the open unit disc and f : D → C be meromorphic and injective in D. We assume that f is holomorphic at zero and has the expansion f(z) = z + ∑(a{sub}nz{sup}n)0(n from 2 to ∞). Especially, we consider f that map D onto a domain whose complement with respect to C is convex. We call these functions concave univalent functions and denote the set of these functions by Co. We prove that the sharp inequalities |a{sub}n| ≥ 1, n ∈ N, are valid for all concave univalent functions. Furthermore, we consider those concave univalent functions which have their pole at a point p∈(0,1) and determine the precise domain of variability for the coefficients a{sub}2 and a{sub}3 for these classes of functions.