2022/05/15 by Nujood M. Alshehri, Alshehri, Nujood M., Zinaida A. Lykova +1 · 1 citation
Mathematics · #30E05 #32F45 #93B36 #93B50 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2205.07306
openalex publication_date 2022/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove a Schwarz lemma for the pentablock. The set P=\(a21, tr A, det A) : A=[aij]i,j=12 ∈ \mathbbB2× 2\ where \mathbbB2× 2 denotes the open unit ball in the space of 2× 2 complex matrices, is called the pentablock. The pentablock is a bounded nonconvex domain in ℂ3 which arises naturally in connection with a certain problem of μ-synthesis. We develop a concrete structure theory for the rational maps from the unit disc \BbbD to the closed pentablock P that map the unit circle \mathbbT to the distinguished boundary bP of P. Such maps are called rational P-inner functions. We give relations between penta-inner functions and inner functions from \BbbD to the symmetrized bidisc. We describe the construction of rational penta-inner functions x = (a, s, p) : \BbbD → P of prescribed degree from the zeroes of a, s and s2-4p. The proof of this theorem is constructive: it gives an algorithm for the construction of a family of such functions x subject to the computation of Fejér-Riesz factorizations of certain non-negative trigonometric functions on the circle. We use properties and the construction of rational P-inner functions to prove a Schwarz lemma for the pentablock.