Lower bounds for self-adjoint Sturm-Liouville operators

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