Boundary rigidity for some classes of meromorphic functions

TL;DR

This paper establishes boundary conditions under which certain classes of meromorphic functions, including generalized Schur, Carathéodory, and Nevanlinna functions, are uniquely determined to be a specific rational function on the unit circle.

Contribution

It introduces new boundary asymptotic conditions that ensure the uniqueness of these classes of functions as rational functions on the unit circle.

Findings

Boundary conditions guarantee functions are identical to a given rational function.

Rigidity results for generalized Carathéodory and Nevanlinna functions.

Extension of boundary rigidity concepts to broader classes of meromorphic functions.

Abstract

Sufficient boundary asymptotic conditions are established for a generalized Schur function $f$ to be identically equal to a given rational function $g$ unimodular on the unit circle. Similar rigidity statements are presented for generalized Carath\'eodory and generalized Nevanlinna functions.