Mapping properties of analytic functions on the disk
  There is a universal constant $0<r_0<1$ with the following property. Suppose that $f$ is an analytic function on the unit disk $\D$, and suppose that there exists a constant $M>0$ so that the Euclidean area, counting multiplicity, of the portion of $f(\D)$ which lies over the disk $D(f(0),M)$, centered at $f(0)$ and of radius $M$, is strictly less than the area of $D(f(0),M)$. Then $f$ must send $r_0\bar{\D}$ into $D(f(0),M)$. This answers a conjecture of Don Marshall. 
Pietro Poggi-Corradini