Geometric properties of domains related to μ-synthesis

Zapałowski, Paweł

doi:10.1016/j.jmaa.2015.04.088

Cited by 4 publications

(3 citation statements)

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“…It is known that the pentablock is polynomially convex and starlike, see [4]. It was shown in [31] that the pentablock P is hyperconvex and that P cannot be exhausted by domains biholomorphic to convex ones. Later in [30,Theorem 1.1] it was proved that P is a C-convex domain.…”

Section: Remark 26mentioning

confidence: 99%

A Schwarz Lemma for the Pentablock

Alshehri

Lykova

2022

J Geom Anal

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In this paper, we prove a Schwarz lemma for the pentablock. The pentablock $$\mathcal {P}$$ P is defined by $$\begin{aligned} \mathcal {P}=\{(a_{21}, {\text {tr}}A, \det A) : A=[a_{ij}]_{i,j=1}^2 \in \mathbb {B}^{2\times 2}\} \end{aligned}$$ P = { ( a 21 , tr A , det A ) : A = [ a ij ] i , j = 1 2 ∈ B 2 × 2 } where $$\mathbb {B}^{2\times 2}$$ B 2 × 2 denotes the open unit ball in the space of $$2\times 2$$ 2 × 2 complex matrices. The pentablock is a bounded non-convex domain in $$\mathbb {C}^3$$ C 3 which arises naturally in connection with a certain problem of $$\mu $$ μ -synthesis. We develop a concrete structure theory for the rational maps from the unit disc $$\mathbb {D}$$ D to the closed pentablock $$\overline{\mathcal {P}}$$ P ¯ that map the unit circle $$\mathbb {T}$$ T to the distinguished boundary $$b\overline{\mathcal {P}}$$ b P ¯ of $$\overline{\mathcal {P}}$$ P ¯ . Such maps are called rational $${\overline{\mathcal {P}}}$$ P ¯ -inner functions. We give relations between $${\overline{\mathcal {P}}}$$ P ¯ -inner functions and inner functions from $$\mathbb {D}$$ D to the symmetrized bidisc. We describe the construction of rational $${\overline{\mathcal {P}}}$$ P ¯ -inner functions $$x = (a, s, p) : \mathbb {D} \rightarrow \overline{\mathcal {P}}$$ x = ( a , s , p ) : D → P ¯ of prescribed degree from the zeroes of a, s and $$s^2-4p$$ s 2 - 4 p . 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 $${\overline{\mathcal {P}}}$$ P ¯ -inner functions to prove a Schwarz lemma for the pentablock.

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Section: Remark 26mentioning

confidence: 99%

A Schwarz Lemma for the Pentablock

Alshehri

Lykova

2022

J Geom Anal

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“…The pentablock is an inhomogeneous domain, see [23]. The complex geometry and function theory of the pentablock were further developed in [6,23,26,27].…”

Section: Introductionmentioning

confidence: 99%

Rational Penta-Inner Functions and the Distinguished Boundary of the Pentablock

Abhay

Kumar²

2022

Complex Anal. Oper. Theory

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“…This new domain is also arising from the μ-synthesis, just like the symmetrized bidisc and the tetrablock. So it is naturally to consider analogous properties of the pentablock, such as the question about C-convexity of P, and Lemperts theorem on the equality of holomorphically invariant functions and metrics for the pentablock (see [1,12,19]). In this paper, we give a positive answer to the C-convexity of P. More precisely, we obtain the following theorem.…”

Section: Introductionmentioning

confidence: 99%