  We prove that the second positive Neumann eigenvalue of a bounded
simply-connected planar domain of a given area does not exceed the first
positive Neumann eigenvalue on a disk of a twice smaller area. This estimate is
sharp and attained by a sequence of domains degenerating to a union of two
identical disks. In particular, this result implies the Polya conjecture for
the second Neumann eigenvalue. The proof is based on a combination of analytic
and topological arguments. As a by-product of our method we obtain an upper
bound on the second eigenvalue for conformally round metrics on odd-dimensional
spheres.
