In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.
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.
Differential (or integral) inequalities, derived from differential (respectively, integral) equations by replacing the equality sign with an inequality sign, form a broad class of such auxiliary relations.
[1][2] One instance of such theorem was used by Aronson and Weinberger to characterize solutions of Fisher's equation, a reaction-diffusion equation.
[3] Other examples of comparison theorems include: In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.