If $f:D\to D’$ is analytic and $u: D'\to R$ is harmonic then the composition of $u$ and $f$ is harmonic in $D$
Solution 1:
Locally every harmonic function $u$ is the real part of an analytic function $g$, so locally $u \circ f = \operatorname{Re} (g \circ f)$ is the real part of an analytic function, hence harmonic. A function which is everywhere locally harmonic is globally harmonic, showing that the statement is true.