Classical convexity inequalities -- Jensen's inequality, the Jensen-Steffensen inequality, and inequalities comparing two Jensen functionals -- are sharpened by inserting explicit correction terms built from a convexity modulus Phi, where a function is uniformly convex when the chord-minus-function gap is bounded below by t(1-t)Phi(|x-y|). Learners see how refinement techniques previously available only for strongly convex or superquadratic functions extend to the broader classes of uniformly convex and Phi-convex functions, producing quantitatively stronger lower bounds for the Jensen differences.
This real-analysis paper strengthens three classical convexity inequalities -- Jensen's inequality, the Jensen-Steffensen inequality, and inequalities comparing two Jensen functionals -- by inserting…