Comparison theorem

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.^[1][2]

• Chaplygin's theorem; Chaplygin inequality^[3]
• Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations.
• Sturm comparison theorem
• Aronson and Weinberger used a comparison theorem to characterize solutions to Fisher's equation, a reaction--diffusion equation.^{[citation needed]}
• Hille-Wintner comparison theorem

In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. ^[4]

• Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart.
• Toponogov's theorem
• Myers's theorem
• Hessian comparison theorem
• Laplacian comparison theorem
• Morse–Schoenberg comparison theorem
• Berger comparison theorem, Rauch–Berger comparison theorem^[5]
• Berger–Kazdan comparison theorem^[6]
• Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold)^[7]
• Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures^[8]
• Lichnerowicz comparison theorem
• Eigenvalue comparison theorem
  • Cheng's eigenvalue comparison theorem
• See also: Comparison triangle

• Limit comparison theorem, about convergence of series
• Comparison theorem for integrals, about convergence of integrals
• Zeeman's comparison theorem, a technical tool from the theory of spectral sequences