Functional differential inequalities and estimation of the Cauchy function of an equation with aftereffect

We consider scalar functional differential inequalities that are used to estimate solutions of differential equations with deviating argument. A theorem on positiveness of the Cauchy function of a differential equation with aftereffect is derived from a theorem on a functional differential inequality with nonlinear monotone operator, which is a direct generalization of the simplest classical theorem on a differential inequality. The suggested proofs rely on local properties of continuous functions, only.