Stieltjes functions of finite order and hyperbolic monotonicity

TL;DR

This paper introduces a new class of finite-type Stieltjes functions satisfying derivative conditions, establishes their unique integral representations, and reveals a correspondence with hyperbolically monotone functions, extending previous representations.

Contribution

It defines finite-type Stieltjes functions with derivative conditions, provides their integral representations, and links them to hyperbolically monotone functions, generalizing earlier results.

Findings

Unique integral representation for finite-type Stieltjes functions.

A two-to-one correspondence between these functions and hyperbolically monotone functions.

Extension of previous representations of HCM functions via Stieltjes transforms.

Abstract

A class of Stieltjes functions of finite type is introduced. These satisfy Widder's conditions on the successive derivatives up to some finite order, and are not necessarily smooth. We show that such functions have a unique integral representation, along some generic kernel which is a truncated Laurent series approximating the standard Stieltjes kernel. We then obtain a two-to-one correspondence, via the logarithmic derivative, between these functions and a subclass of hyperbolically monotone functions of finite type. This correspondence generalizes a representation of HCM functions in terms of two Stieltjes transforms earlier obtained by the first author.