Sharp inequalities for the coefficients of concave schlicht functions

Abstract

Let D denote the open unit disc and let f: D → ℂ be holomorphic and injective in D. We further assume that f(D) is unbounded and ℂ \ f(D) is a convex domain. In this article, we consider the Taylor coefficients a n(f) of the normalized expansion f(z) = z + Σ n=2 ∞an(f)zn, z ∈ D, n=2 and we impose on such functions f the second normalization f(1) = ∞. We call these functions concave schlicht functions, as the image of D is a concave domain. We prove that the sharp inequalities |an(f)-n+1/2 ≤ n-1/2, n≥2, are valid. This settles a conjecture formulated in [2]. © Swiss Mathematical Society