  We show how a large class of sufficient conditions for the existence of bound
states, in non-positive central potentials, can be constructed. These
sufficient conditions yield upper limits on the critical value,
$g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the
potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears.
These upper limits are significantly more stringent than hitherto known
results.
