Real-variables characterization of generalized Stieltjes functions

TL;DR

This paper characterizes generalized Stieltjes functions of any order using derivative inequalities on positive real numbers, providing a new proof for the classical case when the order is one.

Contribution

It offers a novel derivative-based characterization for generalized Stieltjes functions of any order, extending and simplifying previous results.

Findings

Characterization of generalized Stieltjes functions via inequalities

New proof of Widder's characterization for classical Stieltjes functions

Extension of characterization to any order  > 0

Abstract

We obtain a characterization of generalized Stieltjes functions of any order \lambda > 0 in terms of inequalities for their derivatives on (0,\infty). When \lambda=1, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.