Eigenvalue ratios for the conformable fractional vibrating string equations with single-well densities

In this paper, we investigated in a class of Sturm-Liouville vibrating string problems involving conformable fractional derivatives and studied the behavior of their eigenvalue ratios. Under the assumptions that the potential function is of single-barrier type or the density function is of single-well type, we rigorously established the optimal upper bound for the eigenvalue ratios ${\lambda }_n/{\lambda }_m$, namely

$$\frac{{\lambda }_n}{{\lambda }_m}\le {\left(\frac{n}{m}\right)}^2(n>m).$$

Moreover, we showed that equality holds if and only if the density or the potential is constant. Our novelty was in extending the classical Sturm–Liouville eigenvalue inequalities to the conformable fractional setting.