= Spectral-radius maximum principle
{title2=$r(f(z))\le\sup_{w\in\partial K}r(f(w))$}
For a <Banach algebra>-valued <holomorphic function>, set $p_n(w)=\|f(w)^{2^n}\|^{1/2^n}$. These <continuous> functions decrease to the <spectral radius> by the <spectral radius formula>. If $M=\sup_{\partial K}r(f(w))$, the <continuous> functions $\max\{p_n,M\}$ on $\partial K$ decrease to the <continuous> constant $M$. The <Dini theorem> gives <uniform convergence>. Apply the <norm maximum principle for a Banach-space-valued holomorphic function> to each power, and then let $n\to\infty$. This proof does not assume <continuity> of <spectral radius>.
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