= Solution
Take the <Laurent series> of $f$ on $0<|z|<r$. For every integer $k\geq1$, its <coefficient> of $z^{-k}$ satisfies, on a circle of radius $0<\rho<r$,
$$
|c_{-k}|=\left|\frac1{2\pi i}\int_{|\zeta|=\rho}f(\zeta)\zeta^{k-1}\,d\zeta\right|\leq m\rho^k.
$$
The <coefficient> is independent of $\rho$. Letting $\rho\downarrow0$ therefore gives $c_{-k}=0$ for every $k\geq1$. The <Laurent series> is a <power series> with no negative powers and supplies a <holomorphic extension> at zero. Thus \b[zero is a <removable singularity>]. This is the <Riemann removable singularity theorem>, with boundedness used to eliminate the entire principal part.
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