  In this note we provide estimates for the lower bound of the self-adjoint
operator associated with the three-coefficient Sturm-Liouville differential
expression $$ \frac{1}{r} \left(-\frac{\mathrm d}{\mathrm dx} p \frac{\mathrm
d}{\mathrm dx} + q\right) $$ in the weighted $L^2$-Hilbert space $L^2(\mathbb
R; rdx)$.
