Plane harmonicity is preserved by holomorphic and antiholomorphic changes of coordinate

For the holomorphic case, fix $a\in U$. By [L1], choose a neighbourhood $W$ of $\varphi (a)$ and a holomorphic function $F$ on $W$ with $\mathrm{Re}F=u$ there. Shrinking around $a$ if necessary, $\varphi$ maps that neighbourhood into $W$, so [L2] makes $F\circ \varphi$ holomorphic and $\mathrm{Re}(F\circ \varphi )=u\circ \varphi$ there. Thus $u\circ \varphi$ is harmonic near $a$, and since $a$ was arbitrary it is harmonic on $U$.

For the antiholomorphic case, fix $a\in U$ and choose $W$ and $F$ as in step 1.1 around $\psi (a)$. By [L3], the map $\tilde{\psi }(z):=\overline{\psi (z)}$ is holomorphic on $U$, and the map $\tilde{F}(w):=\overline{F(\stackrel{‾}{w})}$ is holomorphic on $\bar{W}:=\{\bar{\zeta }:\zeta \in W\}$. Therefore [L2] makes $\tilde{F}\circ \tilde{\psi }$ holomorphic on a neighbourhood of $a$, and its real part is $\mathrm{Re}\tilde{F}(\tilde{\psi }(z))=\mathrm{Re}\overline{F(\psi (z))}=\mathrm{Re}F(\psi (z))=u(\psi (z))\mathrm{.}$ Hence $u\circ \psi$ is harmonic near $a$, and therefore on $U$.

Steps 1.1 and 2.1 prove the two invariance statements.