One-dimensional Rauch comparison inequality
= One-dimensional Rauch comparison inequality
If $h''+Kh=0$, $h(0)=0$, $h'(0)=1$, $h>0$, and $K\leq C$ for $C>0$, then
$$
h(r)\geq\frac{\sin(\sqrt C r)}{\sqrt C}
\qquad(0\leq r<\pi/\sqrt C).
$$
Subtract a smaller multiple of the sine solution and use the monotonicity of its Wronskian with $h$.