List of Bounds

Inequalities are mysterious yet necessary. Here’s a running list of every inequality I’ve applied in research.

Basics

Grönwall’s Inequality Let \(T>0\) and \(f: [0,T] \to \mathbb R\) be differentiable. Suppose there is a time-dependent constant \(C_t\) \[\begin{align} f'(t) \leq C_t \, f(t), \qquad \forall t \in [0,T] \end{align}\] Then \[\begin{align} f(t) \leq f(0) \exp \left(\int_0^t C_s\, ds \right) \end{align}\]