A useful tool in obtaining bounds for solutions of differential equations is the Gronwall inequality, also known as Bellman’s Lemma, and its various generalizations. The content of these theorems is to compare solutions of certain inequalities with solutions of the corresponding equation. In this note we establish, using a lattice fixed point theorem, a similar result for a functional equation and show that it includes a particularly useful form of Bellman’s Lemma due to Viswanatham. Let [a, 61 denote an interval, possibly infinite but which we will assume closed at any finite endpoint, and let U(X) and p(x) denote functions whose range is in [a, b] whenever the domain is [a, 61. The functional equation which we consider is