Inequalities and higher order convexity

TL;DR

This paper investigates conditions under which weighted inequalities hold for functions with nonnegative higher order derivatives, generalizing classical inequalities like Fuchs and Schur convexity through combinatorial methods.

Contribution

It introduces new conditions for inequalities involving functions with nonnegative kth derivatives, extending classical results with combinatorial proofs.

Findings

Derived generalized inequalities for functions with nonnegative kth derivatives

Established criteria encompassing Fuchs inequality and Schur convexity

Provided combinatorial proofs for the generalized theorems

Abstract

We study the following problem: given n real arguments a1, ..., an and n real weights w1, ..., wn, under what conditions does the inequality w1 f(a1) + w2 f(a2) + ... + wn f(an) >= 0 hold for all functions f with nonnegative kth derivative for some given integer k? Using simple combinatorial techniques, we can prove many generalizations of theorems ranging from the Fuchs inequality to the criterion for Schur convexity.