Let be $\mathcal M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\partial\mathbb D$ (unit circle in the complex plane). Define a mapping $\Phi:\mathcal M(\partial\mathbb D)\to \ell^{\infty}(\mathbb N)$ by
$$
\Phi(\mu)=(c_0,c_1,\ldots)
$$
where $c_k$, are the coefficients of the Taylor expansion (around $z=0$) of the function
$$
f(z)=\int_{\partial\mathbb D}\frac{1}{1-wz} d\mu(w),
$$
**Question:** Who is the set $\Phi(\mathcal M(\partial\mathbb D))$ ? Given a sequence is there a simple criterion to verify if it belongs to this space?

**Motivation:** Let be $A=(a_0,a_1,\ldots)\in\Phi(\mathcal M(\mathbb D))$, $g:U\subset\mathbb C\to \mathbb C$ analytic with
$$g(z)=\sum_{k=0}^{\infty}b_kz^k$$
Define $A*g:U\subset\mathbb C\to \mathbb C$ by
$$
A*h(z)=\sum_{k=0}^{\infty} a_kb_kz^k.
$$
The above integral representation, can be used to give a short proof of
$$
\|A*h\|_{U}\leq \|h\|_{U},
$$
where $\|g\|_{U}=\sup_{z\in U}|g(z)|$.

PS:This question it is a generalization of a question that arose in a discussion at Area 51.

** Edition: ** I am correcting the question, because in the previous version the integrals could have no meaning. In fact, I was looking for the criterion by a line integral representation.