  Let $E$ be the open region in the complex plane bounded by an ellipse. The B.
and F. Delyon norm $\|\cdot\|_{\mathrm{bfd}}$ on the space $\mathrm{Hol}(E)$ of
holomorphic functions on $E$ is defined by $$ \|f\|_{\mathrm{bfd}}
\stackrel{\rm def}{=} \sup_{T\in \mathcal{F}_{\mathrm {bfd}}(E)}\|f(T)\|, $$
where $\mathcal{F}_{\mathrm {bfd}}(E)$ is the class of operators $T$ such that
the closure of the numerical range of $T$ is contained in $E$. The name of the
norm recognizes a celebrated theorem of the brothers Delyon, which implies that
$\|\cdot\|_{\mathrm{bfd}}$ is equivalent to the supremum norm
$\|\cdot\|_\infty$ on $\mathrm{Hol}(E)$.
  The purpose of this paper is to develop the theory of holomorphic functions
of bfd-norm less than or equal to one on $E$. To do so we shall employ a
remarkable connection between the bfd norm on $\mathrm{Hol}(E)$ and the
supremum norm $\|\cdot\|_\infty$ on the space $\mathrm{H}^\infty(G)$ of bounded
holomorphic functions on the symmetrized bidisc, the domain $G$ in
$\mathbb{C}^2$ defined by \begin{align*} G & \stackrel{\rm def}{=} \{(z+w,zw):
|z|<1, |w|<1\}. \end{align*} It transpires that there exists a holomorphic
embedding $\tau:E \to G$ having the property that, for any bounded holomorphic
function $f$ on $E$, \[ \|f\|_{\mathrm{bfd}} = \inf\{\|F\|_\infty: F \in
{\mathrm H}^\infty(G), F\circ\tau=f\}, \] and moreover, the infimum is attained
at some $F \in \mathrm{H}^\infty(G)$. This result allows us to derive, for
holomorphic functions of bfd-norm at most one on $E$, analogs of the well-known
model and realization formulae for Schur-class functions. We also give a second
derivation of these models and realizations, which exploits the Zhukovskii
mapping from an annulus onto $E$.
