[en] In this paper, after first showing that for everywhere attractive potentials, lambda U(r)(<=)o for all r, lambda>0, if U(r) falls off as fast as or faster than r^-2 as r → infinity, the radial Schroedinger equation does or does not have bound state solutions according to whether lambda>lambda₀ or lambda< lambda₀, where lambdasub(deg)>0, an exact expression is derived for the minimum value lambdasub(deg) in terms of the potential only, for potentials U(r) such that rsup(m)U(r)→0 as r→infinity for some m> 3. It is shown that in the process the work improves upon a result that can be deduced from Bargmann's inequality