Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write $$\omega=\partial\bar\partial\log f$$. So, my question is can say $f$ is equal to $h$ up to additional constant? if we have $f$ then how can we find $h$
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The function $\psi:=\log(fh^{1})$ satisfies $\partial\bar\partial \psi=0$, because $ \partial\bar\partial\log f=\partial\bar\partial\log h =\omega$. Such functions are called pluriharmonic. Locally a pluriharmonic function is a real part of a holomorphic function, by PoincareDolbeaultGrothendieck lemma. This fact is true globally, because any realvalued holomorphic function vanishes, and therefore the local holomorphic functions can be glued together. Then, $h$ is unique (up to a constant multiplier) if and only if the manifold has no nonconstant global holomorphic functions. 

