Smooth maps of kahler manifold

Question

Let $(M,g ,J , \omega)$ be a Kahler manifold.
Prove that the following two statements are equivalent foe very smooth functions $f :M \rightarrow \mathbb{C}$.

1. the Hamiltonian vector field $X := df$ preserves metric $g$ i.e $$ L_X g=0$$ where $L_X$ is lie derivative along $X$;
2. denoting by $D$ and $\overline{D}$ the holomorphic and antiholomorphic part of covariant derivative induce by $g$, we have $$ DDf=0 \quad \quad \overline{D}~ \overline{D}f=0 $$.