= Solution
For the nonnegative <Hodge Laplacian> $\Delta_d^{\mathrm H}=dd^*+d^*d$, the <Kähler Laplacian identity> gives $\Delta_d^{\mathrm H}=2\Delta_{\bar\partial}$. If $u$ is a <holomorphic function>, then $\bar\partial u=0$, while $\bar\partial^*u=0$ by degree. Therefore
$$
\Delta_{\bar\partial}u=(\bar\partial\bar\partial^*+\bar\partial^*\bar\partial)u=0,
\qquad \Delta_d^{\mathrm H}u=0.
$$
The divergence-of-gradient convention printed in part (b) is $\Delta_d=-\Delta_d^{\mathrm H}$ on functions. The sign changes no kernel, so the requested <harmonicity of holomorphic functions on a Kähler manifold> is
$$
\boxed{\Delta_du=0.}
$$
\b[Holomorphic functions are harmonic without any compactness assumption.] This is a local identity from the <Kähler Laplacian identity>, not an integration argument requiring a compact manifold.
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