  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.
