Let $f$ be a continuous map of the closure $\overline{\mathrm{\Delta }}$ of the open unit disc $\mathrm{\Delta }$ of $\mathbb{C}$ into a unital associative Banach algebra $\mathcal{A}$, whose restriction to $\mathrm{\Delta }$ is holomorphic, and which satisfies the condition whereby $0\notin \sigma \left(f\left(z\right)\right)\subset \overline{\mathrm{\Delta }}$ for all $z\in \mathrm{\Delta }$ and $\sigma \left(f\left(z\right)\right)\subset \partial \mathrm{\Delta }$ whenever $z\in \partial \mathrm{\Delta }$ (where $\sigma \left(x\right)$ is the spectrum of any $x\in \mathcal{A}$). One of the basic results of the present paper is that $f$ is , that is to say, $\sigma \left(f\left(z\right)\right)$ is then a compact subset of $\partial \mathrm{\Delta }$ that does not depend on $z$ for all $z\in \overline{\mathrm{\Delta }}$. This fact will be applied to holomorphic self-maps of the open unit ball of some $J^{*}$-algebra...