A Moment Problem on Jordan Domains

Abstract

Let ${D_1},{D_2}$ be Jordan domains on the complex z-plane such that ${\smallint _{{D_1}}}{z^n}dm = {\smallint _{{D_2}}}{z^n}dm$ for every nonnegative integer n. Here m denotes two-dimensional Lebesgue measure. Does it follow that ${D_1} = {D_2}$? This moment problem on Jordan domains was posed by H. S. Shapiro [2, p. 193, Problem 1]. In this paper we construct a counterexample and study conditions on ${D_1}$ and ${D_2}$ which imply that the above equality does not hold for some n.