MONOTONICITY OF WEIGHTED AVERAGES OF CONVEX FUNCTIONS

摘要

We consider weighted averages of the form B-n(W, f) = Sigma(n)(r=0) w(n,r)f(r/n), where W is a summability matrix and f is convex. Conditions are given for B-n(W, f) to increase or decrease with n. It decreases whenever W is a Hausdorff mean. The sequence of Bernstein polynomials for a convex function is a special case.