Mapping properties of analytic functions on the unit disk

Abstract

Let f be analytic on the unit disk D with f(0) = 0. In 1989, D. Marshall conjectured the existence of the universal constant r 0 > 0 such that f(r 0 D) C D M := {w: |w < M} whenever the area, counting multiplicity, of a portion of f(D) over D M is < πM 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..., which is sharp for the problem in this larger class but is not sharp for Marshall's problem.