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

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

are valid. This settles a conjecture formulated in [2].