In this talk I will demonstrate inequalities for four different types of inhomogeneous vibrating membranes which were proved by Hersch using conformal techniques inspired by Szego. Specifically, the inequalities are lower bounds on the reciprocal sum of the first 3 non-trivial eigenvalues of the Laplacian of the membranes. It turns out that in each case the bound is achieved by a homogeneous spherical membrane, giving us the four isoperimetric properties promised in the title.