Sharp inequalities for the coefficients of concave schlicht functions

Abstract

Let D denote the open unit disc and let f\colon D\to \mathbb{C} be holomorphic and injective in D . We further assume that f(D) is unbounded and \mathbb{C}\setminus f(D) is a convex domain. In this article, we consider the Taylor coefficients a_n(f) of the normalized expansion f(z)=z+\sum_{n=2}^{\infty}a_n(f)z^n, z\in D, and we impose on such functions f the second normalization f(1)=\infty . We call these functions concave schlicht functions, as the image of D is a concave domain. We prove that the sharp inequalities |a_n(f)-\frac{n+1}{2}|\leq\frac{n-1}{2}, n\geq 2, are valid. This settles a conjecture formulated in [2].