For questions about the use of maximum principle, either in the context of partial differential equations or complex analysis.
In the context of partial differential equations, the maximum principle states that
If $f$ is harmonic, then $f$ cannot attain a local maximum without being constant. That is, if $D$ is a connected open subset of $\mathbb{R}^n$ and there exists $x_0 \in D$ with $$|f(x)| \ge |f(x_0)|$$ for every $x$ in a neighborhood of $x_0$, then $f$ is identically a constant function.
In complex analysis, the maximum principle (usually called the maximum modulus principle) states an analogous result:
If $f$ is holomorphic, then $|f|$ cannot attain a local maximum without being constant. That is, if $D$ is a connected open subset of $\mathbb{C}$ and there exists $z_0 \in D$ with $$|f(z)| \ge |f(z_0)|$$ for every $z$ in a neighborhood of $z_0$, then $f$ is identically a constant function.
References: Maximum principle (PDE), maximum modulus principle.
