Concave univalent functions and Dirichlet finite integral

Abstract

The article deals with the class ${\mathcal F}_{\alpha }$ consisting of non-vanishing functions $f$ that are analytic and univalent in \ID such that the complement \IC\backslash f(\ID) is a convex set, $f(1)=\infty ,$ $f(0)=1$ and the angle at $\infty$ is less than or equal to $\alpha \pi ,$ for some $\alpha \in (1,2]$. Related to this class is the class $CO(\alpha)$ of concave univalent mappings in \ID, but this differs from ${\mathcal F}_{\alpha }$ with the standard normalization $f(0)=0=f'(0)=1.$ A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for $CO(2)$ settled by Avkhadiev et al. \cite{Avk-Wir-04}. Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for $CO(\alpha)$ is also presented.Comment: 15 pages; A version of it will appear in the journal "Mathematische Nachrichten