Concave univalent functions and Dirichlet finite integral

TL;DR

This paper studies classes of concave univalent functions, exploring their properties, proving a coefficient conjecture for a specific case, and examining the Yamashita conjecture related to Dirichlet integrals.

Contribution

It introduces and analyzes properties of the class ${\

Findings

Proof of the coefficient conjecture for $CO(2)$.

Discussion of properties of ${\mathcal F}_{\alpha}$ and $CO(\alpha)$.

Results related to the Yamashita conjecture on 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.