A note on the eigenvalue ratio of vibrating strings (Q1046898)

This note deals with the ratio \(\lambda_2/\lambda_1\) of the first two eigenvalues of the Dirichlet problem on \([0,a]\) for equations of the form \[ u^{\prime\prime}(x)+\lambda \rho(x) u(x)=0, \] where \(\rho\) is continuous, symmetric, and nonnegative. The assumption in the author's earlier paper [Proc. Am. Math. Soc. 127, No.~6, 1805--1813 (1999; Zbl 0917.34067)] that \(\rho\) should possess a single minimum is slightly relaxed here.