1. This note is concerned with inequalities of the form(1.1)where the open interval (a,b) of integration may be bounded or unbounded, i.e. −∞ ⋚ a ⋚ b ⋚ ∞, the coefficients p, q and w are real-valued on (a,b), w is non-negative, M is the symmetric differential expression(1.2)and η is the largest linear manifold of real-valued functions on (a,b), so chosen that the integrals on the right-hand side are both finite.