Eigenvalue ratios for vibrating String equations with concave densities

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.