``Bounds on the product of the first two non-trivial frequences of a free membrane'' In this talk we are interested in the eigenvalue problem of a free membrane represented as a bounded simply-connected planar domain D with Lipschitz boundary. The aim of this talk is twofold. First, we give a positive answer to the following conjecture of Iosif Polterovich: the product of the first two non-trivial Neumann eigenvalues of the laplacian on D (frequencies of the free membrane D) is upper bounded by the value of the same quantity for the disk with same area as D. This estimate is sharp and the equality occurs if and only if D is a disk. Secondly, we consider the class of n-sided convex polygons and establish an isoperimetric inequality for the product of some moments of inertia. As an application, we obtain an explicit nice upper bound for the product of the first two non-trivial frequences of a free membrane represented as a n-sided convex polygon.