Zeros of nonpositive type of generalized Nevanlinna functions with one negative square

TL;DR

This paper investigates the properties of zeros of nonpositive type for generalized Nevanlinna functions with one negative square, analyzing how these zeros behave under fractional linear transformations and defining a path in the complex plane.

Contribution

It characterizes the behavior of the generalized zero of nonpositive type under fractional linear transformations for functions with one negative square.

Findings

The zero of nonpositive type is unique in the closed extended upper halfplane.

The path of zeros under transformations is well-defined and studied in detail.

Properties of the zero path are systematically analyzed.

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 $Q_\tau(z)=(Q(z)-\tau)/(1+\tau Q(z))$, $\tau \in \mathbb{R} \cup \{\infty\}$, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type $\alpha(\tau)$ as a function of $\tau$ defines a path in the closed upper halfplane. Various properties of this path are studied in detail.