Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 7 / 4 / Solution

Apply the spectral-radius maximum principle to the entire Banach-space-valued holomorphic function $f(z)=a+zb$ on the closed disk $|z|\le \epsilon$. On its boundary the assumed estimate gives $r(f(z))\le C\epsilon$. Thereforefor every $\epsilon >0$. Letting $\epsilon \downarrow 0$ provesThe puncture at $z=0$ in the hypothesis causes no difficulty: the bound is used only on the nonzero boundary circle.