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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects. In Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in 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.