Kolmogorov's Problem on the Class of Multiply Monotone Functions

TL;DR

This paper establishes necessary and sufficient conditions for the existence of multiply monotone functions on the negative half-line with prescribed uniform norms of derivatives, advancing the understanding of Kolmogorov's problem in this context.

Contribution

It provides a complete characterization of when a set of derivative norms can be realized by an r-monotone function on the negative half-line.

Findings

Derived necessary and sufficient conditions for the problem.

Extended Kolmogorov's problem to multiply monotone functions.

Clarified the structure of functions meeting the norm constraints.

Abstract

In this paper we give necessary and sufficient conditions for the system of positive numbers $ M_{k_1}, M_{k_2},..., M_{k_{d}},$ $0\leq k_1<...<k_{d} {\leq} r$, to guarantee the existence of an $r$-monotone function defined on the negative half-line $\RR_-$ and such that $\|x^{(k_i)}\|_{\infty}=M_{k_i}, i=1,2,...,d$.