Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.