A Schwarz Lemma for the Pentablock

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