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A Schwarz Lemma for the Pentablock.
Nujood M Alshehri1, Zinaida A Lykova1
1School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU UK.
Journal of Geometric Analysis
|May 22, 2026
Summary
This study introduces a Schwarz lemma for the pentablock, a complex domain relevant to μ-synthesis. Researchers developed a construction theory for rational maps, proving a Schwarz lemma for this domain.
Area of Science:
- Complex Analysis
- Operator Theory
- Control Theory
Background:
- The pentablock is a bounded, non-convex domain in C^3, arising from μ-synthesis problems.
- Rational maps and inner functions are crucial in complex analysis and systems theory.
- Schwarz lemmas provide fundamental inequalities for analytic maps between domains.
Purpose of the Study:
- To prove a Schwarz lemma for the pentablock domain.
- To develop a structure theory for rational maps into the pentablock.
- To establish connections between pentablock-inner functions and inner functions of the symmetrized bidisc.
Main Methods:
- Definition of the pentablock P as the image of 2x2 complex matrices in the unit ball under mapping to (a21, trA, detA).
- Development of a structure theory for rational maps from the unit disc D to the closed pentablock P̄.
- Construction of rational P̄-inner functions using zeroes and Fejér-Riesz factorizations.
Main Results:
- A concrete structure theory for rational maps from D to P̄ that map the unit circle T to the boundary bP̄ is established.
- Relationships between P̄-inner functions and inner functions of the symmetrized bidisc are identified.
- A constructive method for generating rational P̄-inner functions of prescribed degree is presented.
Conclusions:
- The study successfully proves a Schwarz lemma for the pentablock.
- The developed constructive theory provides an algorithmic approach for creating specific rational inner functions.
- These findings advance the understanding of complex domains and their associated mappings in mathematical analysis and engineering.
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