On quasi-Herglotz functions in one variable

TL;DR

This paper introduces and characterizes quasi-Herglotz functions, expanding the class of Herglotz functions, and explores their subclasses and connections to other mathematical areas.

Contribution

It defines quasi-Herglotz functions as a new class, provides analytic characterizations, and investigates special subclasses and their relations to existing theories.

Findings

Characterization theorems for quasi-Herglotz functions

Analysis of zero in half-plane subclasses

Investigation of rational quasi-Herglotz functions

Abstract

In this paper, the class of (complex) quasi-Herglotz functions is introduced as the complex vector space generated by the convex cone of ordinary Herglotz functions. We prove characterization theorems, in particular, an analytic characterization. The subclasses of quasi-Herglotz functions that are identically zero in one half-plane as well as rational quasi-Herglotz functions are investigated in detail. Moreover, we relate to other areas such as weighted Hardy spaces, definitizable functions, the Cauchy transform on the unit circle and sum-rule identities.