The set P = {(a 21 , tr A, det A) : A = [a ij ] 2 i,j=1 ∈ B 2×2 } where B 2×2 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 C 3 , and is polynomially convex and starlike. We develop an explicit structure theory for the rational maps from the unit disc D to the closed pentablock P that map the unit circle T 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 D to the symmetrized bidisc. We describe the construction of rational penta-inner functions x = (a, s, p) : D → 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 P-inner functions to prove a Schwarz lemma for the pentablock.

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.