Logarithmically convex domain

Definition

A domain (connected open subset) in $\mathbb{C}$ is termed logarithmically convex or log-convex if we can choose a branch of the logarithm that is well-defined on this domain, and under which the image of this domain is a convex subset of $\mathbb{C}$.

Facts

• Disc away from origin is logarithmically convex: Any disc away from the origin (i.e. not containing the origin in either the boundary or the interior) is logarithmically convex.