Moments and the Range of the Derivative

Abstract

In this note we introduce three problems related to the topic of finite Hausdorff moments. Generally speaking, given the first \(n+1\) (\(n\in \mathbb N\cup \{0\}\)) 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 \(\frac{df}{dx}\)? 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.