  The paper deals with an eigenvalue problems possessing infinitely many
positive and negative eigenvalues. Inequalities for the smallest positive and
the largest negative eigenvalues, which have the same properties as the
fundamental frequency, are derived. The main question is whether or not the
classical isoperimetric inequalities for the fundamental frequency of membranes
hold in this case. The arguments are based on the harmonic transplantation for
the global results and the shape derivatives (domain variations) for nearly
circular domain.
