  In this paper, we prove the optimal lower bound
$\frac{\lambda_n}{\lambda_m}\geq(\frac{n}{m})^2$ of vibrating string
$$-y''=\lambda\rho(x) y,$$ with Dirichlet boundary conditions for concave
densities. Our aproach is based on the method of Huang [Proc. AMS., 1999]. The
main argument is to restrict the two consecutive eigenfunction $y_{n-1}$ and
$y_n$ between two successive zeros of $y_{n-1}$. We also prove the same result
for the Dirichlet Sturm-Liouville problems.

Spectral Theory