Inequality for 3-convex and 4-convex functions

Abstract The inequality for 3-convex functions is established, which is a generalization of Levinson inequality and can unify some Ky Fan type inequalities, power mean inequality can also be refined. The inequality for 4-convex functions is also established, Ky Fan type inequality is refined on a deeper layer and some other special cases are also mentioned. As the obvious advantage of these basic inequalities for higher-order-convex function, Jensen inequality for convex function can be deduced from both inequality for 3-convex function and inequality for 4-convex function. Our proof is based on a lemma for convex function and detailed analysis for integral of derivative function.