# Inverse problem for negative moments

Let $D$ be a bounded simply connected domain in the plane, bounded by a smooth closed curve $\partial D$. Moreover$D$ contains the origin. Assume that all negative complex moments w.r.t arc-length measure are zero. That is

$$\int_{\partial D}z^n ds_z=0\,\,\,,n=-1,-2,\cdots.$$

$\textbf{1.}$ To what extend $D$ can be determined uniquely? In a sense, this problem is similar to F. & M. Riesz theorem but posed for negative moments.

$\textbf{2.}$ Trivially, for $\partial D=\mathbb{S}^1$, all the negative moments are zero. This question is related to another question,see here, that I had posted some time ago which is still unanswered .

$\textbf{3.}$ One can think about the truncated problem, meaning for some $n_0\in\mathbb N$, all the moments with exponent $-n,(-n_0-1),(-n_0-2),\cdots$ vanish. What would be answer to the previous problem?

$\textbf{4.}$ Assuming $\Gamma$ is a smooth closed simple curve, does $$\int_{\partial D}z^n d\mu(z)=0\,\,\,,n=+1,+2,\cdots,$$ where $\mu$ is absolutely continuous to the arc-length measure with $\frac{d\mu}{ds}\in C(\Gamma)$, imply that $\Gamma$ must be a circle?