Let $M_a(\mathbb{C})$ the space of all symmetric (w.r.t conjugation) probability measures $\mu$ on $\mathbb{C}$ such that the support of $\mu$ is included in $R_a:=\{z\in\mathbb{C};\ \Re(z)\leq a\}$, and where $a\in \mathbb{R}$. Define: $\begin{equation} I(\mu):=\int_{\mathbb{C}}|z|^2\mu(dz)-\int_{\mathbb{C}}\int_{\mathbb{C}} \ln|z-z'|\mu(dz)\mu(dz') \end{equation}$ The problem is the following : $\begin{equation} \mbox{Find }\Theta(a)=\displaystyle{\inf_{\mu \in M_a(\mathbb{C})}} I(\mu) \end{equation}$
What I know is that when $a\geq 1$, then the minimizer is the uniform density on the disk of radius 1, and I am interested in what happens for $a<1$.
Thank you for your help!