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].