We introduce a comparison principle for positive supersolutions and subsolutions to the Lane-Emden equation for the p−Laplacian. Then we apply such a comparison principle to obtain a variety of results, as “hierarchy” of solutions, sharp pointwise double-side estimates for positive solutions in convex sets, and a sharp geometric estimate on the generalized principal frequencies of convex sets, which provides a generalization of the so-called Hersch-Protter inequality.