  It is shown that for the Calogero-Cohn type upper bounds on the number of
bound states of a negative spherically symmetric potential $V(r)$, in each
angular momentum state, that is, bounds containing only the integral
$\int^\infty_0 |V(r)|^{1/2}dr$, the condition $V'(r) \geq 0$ is not necessary,
and can be replaced by the less stringent condition $(d/dr)[r^{1-2p}(-V)^{1-p}]
\leq 0, 1/2 \leq p < 1$, which allows oscillations in the potential. The
constants in the bounds are accordingly modified, depend on $p$ and $\ell$, and
tend to the standard value for $p = 1/2$.
