Moments and the Range of the Derivative

TL;DR

This paper investigates the relationship between a function's moments and the possible range of its derivative, providing bounds and conjectures based on finite Hausdorff moments and Bernstein polynomial representations.

Contribution

Introduces new bounds on the derivative's range given moments, with explicit results for up to four moments and conjectures for the general case.

Findings

Range of derivative contains convex hull of specific points derived from moments

Explicit solutions for cases with up to three or four moments

Conjectures on the sharpness of the derivative's range bounds

Abstract

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.