= Solution
The local <maximum modulus principle> says that if $|f|$ has a local maximum at an interior point of a connected domain, then $f$ is constant. On a small circle about the maximum, the mean-value property and
$$
|f(z_0)|\le\frac1{2\pi}\int|f(z_0+re^{it})|dt\le|f(z_0)|
$$
force equality everywhere. Equality in the triangle inequality makes the boundary values identical; Cauchy's formula then makes $f$ constant locally, and the identity theorem makes it constant on the domain.
Write $f=u+iv$. The hypothesis gives $u-v\le0$. Therefore
$$
|e^{(1+i)f(z)}|=e^{u-v}\le1,
$$
with equality at zero. The local maximum <modulus> principle makes the exponential constant. <Differentiation> then gives $f'=0$, so $f$ is constant; since $f(0)=0$, it is identically zero.
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