A notion of generalized zero with respect to a linear differential operator $L_n $ for a function f at a singular point of the operator was introduced by Levin and further considered by Willett. This involved a comparison of f with certain solutions of $L_n y = 0$ near the singular point. It is shown that the role of these solutions may be fulfilled by certain solutions of inequalities $L_n y \geqq 0( \leqq 0)$ introduced independently by Hartman and Levin. This result is applied via a generalization of the Polya mean value theorem to the problem of finding best possible relationships between bounds on differential operators and a discussion of the extremals of these relationships. Second order operators are considered in some detail; an analogue of Landau’s inequality is proved for second order operators in which the coefficients need not be constant.