When is the image of a convex set under a holomorphic function convex?

Question

I have a holomorphic function on a domain $G$ containing a compact, convex set $K$. Clearly, if $K$ is a line segment, then $f(K)$ might be an arc on a circle and therefore not convex. But what if the interior of $K$ is non-empty? Is it then true that $f(K)$ is convex?

Answer 1

No. The map $z \mapsto z^2$ sends rectangles to nonconvex curvilinear quadrilaterals with parabolic arcs.