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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 concavedensities. Our aproach is based on the method of Huang [Proc. AMS., 1999]. Themain 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 resultfor the Dirichlet Sturm-Liouville problems.

Eigenvalue ratios for vibrating String equations with concave densities