Inequalities and higher order convexity

Abstract

We study the following problem: given n real arguments a 1 , ..., a n and n real weights w 1 , ..., w n , under what conditions does the inequalityhold for all functions f satisfying f (k) ≥ 0 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.