  In this paper we prove the optimal upper bound
$\frac{\lambda_{n}}{\lambda_{m}}\leq\frac{n^{2}}{m^{2}}$
$\Big(\lambda_{n}>\lambda_{m}\geq 11\sup\limits_{x\in[0,1]}q(x)\Big)$ for
one-dimensional Schrodinger operators with a nonnegative differentiable and
single-barrier potential $q(x)$, such that $\mid q'(x) \mid\leq q^{*},$ where
$q^{*}=\frac{2}{15}\min\{q(0) , q(1)\}$. In particular, if $q(x)$ satisfies the
additional condition $\sup\limits_{x\in[0,1]}q(x)\leq \frac{\pi^{2}}{11}$, then
$\frac{\lambda_{n}}{\lambda_{m}}\leq \frac{n^{2}% }{m^{2}}$ for $n>m\geq 1.$
For this result, we develop a new approach to study the monotonicity of the
modified Pr\"{u}fer angle function.
