 Solution

For $n\ge 0$, putThe bound $r(u)\le \Vert u\Vert$ makes $M$ finite. Each $p_n$ is continuous; submultiplicativity gives $p_{n+1}\le p_n$, and the spectral radius formula gives $p_n(w)\to r(f(w))$. Spectral radius need not be continuous, so applying the Dini theorem directly to $p_n$ would not be justified. Instead define, on the compact boundary,These are continuous, decrease pointwise to the continuous constant $M$, and therefore converge uniformly by the Dini theorem. In particular,Multiplication is a continuous bilinear operation in a Banach algebra, so the product rule shows that $w\mapsto f(w{)}^{2^n}$ is a Banach-space-valued holomorphic function. Part (i), applied to this power, givesTaking the limit on the left and the limit superior on the right proves the spectral-radius maximum principle: