Just before 2.37 Corollary (Spectral Mapping Theorem) Douglas says:
If $\varphi (z)= \sum_{n=0}^\infty a_nz^n$ is an entire function with complex coefficients and $f$ is an element of the Banach algebra $\mathfrak B$, then we let $\varphi (f)$ denote the element $\sum_{n=0}^\infty a_nf^n$ of $\mathfrak B$.
I'm trying to proof that $\sum_{n=0}^\infty a_nf^n$ has sense in $\mathfrak B$ by showing that $\sum_{n=0}^\infty \|a_nf^n\|$ is convergent in $\Bbb C$, but got stucked with the fact that I can't see how $\|f^n\|$ should be bounded. Any help would be appreciated.