An eigenvalue problem for vibrating strings with concave densities

In this paper, we study the order of the Dirichlet and Neumann eigenvalues for the string equation with concave densities. In general, the order of the k-th Dirichlet eigenvalue and (k+1)-th Neumann eigenvalue is indeterminate. But, if the density function ρ is concave on (0,1), we show that the second Neumann eigenvalue is greater than or equal to the first Dirichlet eigenvalue; the equality holds if and only if ρ is constant.