= Solution
Suppose $|g|$ has a local maximum at $z_0$. If $g(z_0)=0$, the nonnegativity of the modulus forces $g=0$ throughout a neighbourhood, and the <identity theorem for holomorphic functions> makes it zero on that connected component.
If $g(z_0)\ne0$, take $c=2i/g(z_0)$. Then $cg(z_0)=2i$, and continuity gives a disk $D$ around $z_0$ on which $\Im(cg)>1$. This lies within the given half-plane for a branch of the <holomorphic logarithm>. On $D$ define
$$
f=\operatorname{Log}(cg),\qquad\Re f=\log|cg|=\log|c|+\log|g|.
$$
Composition preserves holomorphy, and the real logarithm is strictly increasing, so $\Re f$ has a local maximum at $z_0$. Part (i) forces $f$ to be constant on the disk. Exponentiating gives $cg=e^f$, hence $g$ is constant there and, by the <identity theorem for holomorphic functions>, on the whole connected component. Therefore
$$
\boxed{\text{A local maximum of }|g|\text{ forces }g\text{ to be constant on its connected component.}}
$$
This proves the <maximum modulus principle> by the requested logarithm argument. Multiplication by $c$ is what puts the image in the supplied logarithm branch; no global logarithm of $g$ is assumed.
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