Moments and the Range of the Derivative

Eugen J. Ionascu, Richard Stephens

Published 2011-05-06Version 1

In this note we introduce three problems related to the topic of finite Hausdorff moments. Generally speaking, given the first n+1 (n in N or n=0) moments, alpha(0), alpha(1),..., alpha(n), of a real-valued continuously differentiable function f defined on [0,1], what can be said about the size of the image of df/dx? We make the questions more precise and we give answers in the cases of three or fewer moments and in some cases for four moments. In the general situation of n+1 moments, we show that the range of the derivative should contain the convex hull of a set of n numbers calculated in terms of the Bernstein polynomials, x^k(1-x)^{n+1-k}, k=1,2,...,n, which turn out to involve expressions just in terms of the given moments alpha(i), i=0,1,2,...n. In the end we make some conjectures about what may be true in terms of the sharpness of the interval range mentioned before.