A Schwarz Lemma for the Pentablock

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