The von Misis distribution has the highest entropy for a given first circular moment. It seems that the converse is true: for a fixed entropy, the von Misis distribution has the highest first circular moment (in magnitude). This seems like something that should already be proven somewhere, but I haven't been able to find it, nor have my efforts at proving it met with success so far. Does anyone know where a proof of this is (or failing that, how to prove it)?