In the context of a self-adjoint generalized differential system that is equivalent to a type of linear vector Riemann–Stieltjes integral equation, certain functional inequalities are presented generalizing, in particular, the well-known Liapunov inequality $\int_a^b q^ + (t)dt > {4 / {(b - a)}}$, which is satisfied by $q^ + (t) = \frac{1}{2}[q(t) + | {q(t)} |]$ whenever $q(t)$ is a real-valued Lebesgue integrable function on the compact real interval $[a,b]$ which is such that the differential equation $u''(t) + q(t)u(t) = 0$ is oscillatory on $[a,b]$. In particular, some decided extensions of the results of the author's recent paper [19] are given.