If $U\to \mathbb{C}$ is holomorphic and $h:f(U)\to \mathbb{R}$ is harmonic,
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then $h\circ f$ is harmonic.
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Furthermore, Suris gave us a proof with, what he said, was a minor mistake.

Let $z_0\in U$. On an open neighborhood of $f(z_0)$, say $B_{z_0}$, we have that $H:B\to \mathbb{C}$ is holomorphic by the poincare lemma. Here $h$ is the real part of $H$, and we use 7. Proposition 2.1.1. But we can check that $h\circ f$ is the real part of $H\circ f$, and by 6. Corollary 2.1.1 $h\circ f$ must be harmonic.