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.



Coefficients of holomorphic functions defined by Borel probability measures on the unit disc
There's probably something I do not understand about your question, in case just forget my babbling. Anyway let me try: if you expand
$$\frac{1}{1-wz}=\sum\_{k\ge0}z^kw^k$$
and write
$$f(z)=\sum\_{k\ge0}z^k\int\_{\partial D}w^k d\mu(w)=
\sum\_{k\ge0}z^k\int\_0^{2\pi}e^{ik\theta}d\mu(\theta)$$
you see that the $c\_k$ are the Fourier coefficients of positive order of the measure $\mu$. Now I would not go too far and say that any bounded sequence may apply, but this is well documented in several places (see e.g. the [trigonometric moment problem](http://en.wikipedia.org/wiki/Trigonometric_moment_problem)).


EDIT: and maybe [this](https://facultystaff.richmond.edu/~wross/PDF/cauchy.pdf) is a good starting point.


