Global and local behavior of zeros of nonpositive type

TL;DR

This paper investigates the behavior of zeros of nonpositive type in generalized Nevanlinna functions, focusing on their continuity and behavior under fractional linear transformations.

Contribution

It provides a detailed analysis of the continuity and boundary behavior of the generalized zero of nonpositive type for a class of functions with one negative square.

Findings

The zero of nonpositive type varies continuously with the parameter.

Behavior at points where the function extends through the real line is characterized.

The fractional linear transformation preserves the negative square property.

Abstract

A generalized Nevanlinna function $Q(z)$ with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by $Q_\tau(z)=(Q(z)-\tau)/(1+\tau Q(z))$, $\tau \in \mathbf{Real} \cup \{\infty\}$, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type $\alpha(\tau)$ as a function of $\tau$ is being studied. In particular, it is shown that it is continuous and its behavior in the points where the function extends through the real line is investigated.