Mapping properties of analytic functions on the disk

Abstract

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