  In this paper, we prove the optimal upper bound
$\frac{\lambda_n}{\lambda_m}\leq(\frac{n}{m})^2$ of vibrating string
$$-y''=\lambda\rho(x) y,$$ with Dirichlet boundary conditions for single-well
densities. The proof is based on the inequality
$\frac{\lambda_n(\rho)}{\lambda_{m}(\rho)}\leq
\frac{\lambda_n(L)}{\lambda_{m}(L)} ,$ with $L$ must be a stepfunction. We also
prove the same result for the Dirichlet Sturm-Liouville problems.
