\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
It suffices to prove this for maps with domain \(D\subset \CC^n\), and \(f: D\to \CC\). Write \(f=u+\jmath v\). In a small neighborhood of the point \(z\), we can write \(f=\exp(h)\).
Observe that
\[
\Delta \ln|f|=&\Delta \ln|\exp(h)|=u\\
=& \sum_{i=1}^n \partial_{x_i}^2 u+ \partial_{y_i}^2 u\\
=& \sum_{i=1}^n \partial_{x_i}\partial_{y_i}v - \partial_{y_i}\partial_{x_i} v =0
\]
Therefore \(\ln|f|\) is a harmonic function.
We now prove the weak maximum principle for harmonic functions. Let \(h: U\subset \RR^n\to \RR\) be a harmonic function. For \(s\in \RR_{>0}\) we construct the function \(h+s\cdot e^{x_1}\). This now satisfies the property that \(\Delta(h+e^{x_1})>0\) everywhere, and not all of the