  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.
