  Let M be a bounded open plane domain. Let f be a continuous function on the
closure of M, 3-times continuously differentiable in M, which vanish on the
boundary. Polterovich and Sodin proved that the values of f cannot exceed the
norm of the hessian of f, averaged over the entire domain M. In this paper we
study the equality case for this inequality. We show that equality holds if and
only if M is a open disc and f belongs to a special class of radial functions.
