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Let $M_a(\mathbf{C})$ the space of all symmetric (w.r.t conjugation) probability measures $\mu$ on $\mathbf{C}$ such that the support of $\mu$ is included in $R_a:=\{z\in\mathbf{C};\ Re(z)\leq a\}$, and where $a\in \mathbb{R}$. Define: \begin{equation} I(\mu):=\int_{\mathbf{C}}|z|^2\mu(dz)-\int_{\mathbf{C}}\int_{\mathbf{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(\mathbf{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!

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