On the Jensen-Steffensen inequality for generalized convex functions

Jensen-Steffensen type inequalities for P-convex functions and functions with nondecreasing increments are presented. The obtained results are used to prove a generalization of Čebyšev’s inequality and several variants of Hölder’s inequality with weights satisfying the conditions as in the Jensen-Steffensen inequality. A few well-known inequalities for quasi-arithmetic means are generalized.