2024/03/24 by Isabelle Chalendar, Chalendar, I., J. R. Partington +1
Materials Science · Physics and Astronomy · #30D05 #30H10 #94A12 #Advanced X-ray Imaging Techniques #Complex Variables (math.CV) #Electron and X-Ray Spectroscopy Techniques #FOS: Mathematics #Functional Analysis (math.FA) #X-ray Spectroscopy and Fluorescence Analysis
paper · pdf · doi:10.48550/arxiv.2403.16255
openalex publication_date 2024/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f and g be analytic functions on the open unit disc \mathbb D such that |f|=|g| on a set A. We give an alternative proof of the result of Perez that there exists c in the unit circle \mathbb T such that f=cg when A is the union of two lines in \mathbb D intersecting at an angle that is an irrational multiple of π, and from this deduce a sequential generalization of the result. Similarly, the same conclusion is valid when f and g are in the Nevanlinna class and A is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyse the case A=r\mathbb T. Finally, we examine the most general situation when there is equality on two distinct circles in the disc, providing a result or counterexample for each possible configuration.