A comparison theorem (Q5902727)

The principal result in this note is the following: If \[ \int^{(b/2)+h}_{(b/2)-h}[p(x)-(\pi^ 2/b^ 2)]dx\geq 0 \] for each h, \(0<h\leq b/2\), a nonnull solution y(x) of \(y''+p(x)y=0\), \(y(0)=0\), must have a zero \(x=c\) on (0,b] with \(c<b\) unless \(p(x)\equiv \pi^ 2/b^ 2\).