Mapping properties of analytic functions on the unit disk

Let $f$ be analytic on the unit disk $\mathbb{D}$ with $f(0)=0$. In 1989, D. Marshall conjectured the existence of the universal constant $r_0>0$ such that $f(r_0\mathbb{D})\subset \mathbb{D}_M:=\{w:\,\vert w\vert<M\}$ whenever the area, counting multiplicity, of a portion of $f(\mathbb{D})$ over $\mathbb{D}_M$ is $<\pi M^2$. Recently, P. Poggi-Corradini (2007) proved this conjecture with an unspecified constant by the method of extremal metrics. In this note we show that such a universal constant $r_0$ exists for a much larger class consisting of analytic functions omitting two values of a certain doubly-sheeted Riemann surface. We also find a numerical value, $r_0=.03949\ldots$, which is sharp for the problem in this larger class but is not sharp for Marshall's problem.