Comparison theorem

A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects. A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects. In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof) provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property. See also Lyapunov comparison principle In Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.