Sharp inequalities for the coefficients of concave schlicht functions
Summary: 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 by the authors in [Math. Nachr. 271, 3--9 (2004; Zbl 1149.30013)].
scientific article; zbMATH DE number 5071521