On a generalization of a theorem of Levin and Ste\v{c}kin and inequalities of the Hermite-Hadamard type

TL;DR

This paper extends the Levin-Stein theorem by providing new conditions for higher order convex ordering, which can enhance the understanding of Hermite-Hadamard inequalities and quadrature operator inequalities.

Contribution

It introduces generalized necessary and sufficient conditions for higher order convex ordering, broadening the scope of the classical Levin-Stein theorem.

Findings

Generalized convex ordering conditions

Enhanced Hermite-Hadamard inequalities

Applications to quadrature operator inequalities

Abstract

We give new necessary and sufficient conditions for higher order convex ordering. These results generalize the Levin-Ste\v{c}kin theorem (1960) on convex ordering. The obtained results can be useful in the study of the Hermite-Hadamard type inequalities and in particular inequalities between the quadrature operators.