pr.probability - Coefficients of holomorphic functions defined by Borel probability measures on the unit disc

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

Motivation: Let be $A=(a_0,a_1,ldots)inPhi(mathcal M(mathbb D))$, $g:Usubsetmathbb Cto mathbb C$ analytic with
$$g(z)=sum_{k=0}^{infty}b_kz^k$$
Define $A*g:Usubsetmathbb Cto mathbb C$ 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 $|g|_{U}=sup_{zin U}|g(z)|$.

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.