# Geometric properties of measures related to holomorphic functions having positive imaginary or real part

## TL;DR

This paper investigates the geometric and structural properties of measures associated with holomorphic functions that have positive real or imaginary parts, revealing restrictions on their support and simplified forms on hyperplanes.

## Contribution

It characterizes the geometric restrictions and simplifications of measures related to Herglotz-Nevanlinna functions, especially on hyperplanes and within certain regions.

## Key findings

- Restrictions to hyperplanes are simple in form.
- Supports cannot lie within certain geometric regions like positive slope strips.
- Results extend to measures on the unit poly-torus with specific Fourier properties.

## Abstract

In this paper, we study the properties of a certain class of Borel measures on $\mathbb{R}^n$ that arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we find that restrictions to certain hyperplanes are of a surprisingly simple form and show that the supports of such measures can not lie within particular geometric regions, e.g. strips with positive slope. Corresponding results are derived for measures on the unit poly-torus with vanishing mixed Fourier coefficients. These measures are closely related to functions mapping the unit polydisk analytically into the right half-plane.