Sharp inequalities for the coefficients of concave schlicht functions

• Autores: F.G. Aukhadiev, Christian Pommerenke , Karl Joachim Wirths
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 81, Nº 4, 2006, págs. 801-807
• Idioma: inglés
• DOI: 10.4171/cmh/74
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• Resumen

  • 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].