$\partial\bar \partial \log f=\bar \partial \partial \log\bar f$?

Question

Assume $f$ is nonzero holomorphic function. Then is it true $$\partial\bar \partial \log f=\bar \partial \partial \log\bar f?$$ This is one step of construction of Fubini-Study metric in Griffith and Harris's book.

Answer 1

Yes, since they are both zero: $\log f$ is holomorphic, so $\bar\partial \log f = 0$. $\log \bar f$ is anti-holomorphic, so $\partial \log \bar f = 0$. Thus

$$\partial \bar\partial \log f = \bar\partial \partial \log \bar f = 0.$$