In mathematics, Gronwall’s inequality (also called Grönwall’s lemma, Gronwall’s lemma or Gronwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. The differential form was proven by Grönwall in 1919. The integral form was proven by Richard Bellman in 1943. A nonlinear generalization of the Gronwall–Bellman inequality is known as Bihari’s inequality.

First, we consider the Gronwall inequality.

**Type 1**. Bounds by integrals based on lower bound .

Let and be real-valued continuous functions defined on . If is differentiable in and satisfies the differential inequality

then is bounded by the solution of the corresponding differential equation , that is to say

for all .