Let $f$ be a nowhere vanishing holomorphic function some simply connected open subset of $\mathbb{C}^n$.
Let $\partial,\overline{\partial}$ denote the usual Dolbeaut operators.
A book I'm reading seems to claim that $\partial\overline{\partial}\log(|f|^2)$ is always equal to 0. In my attempt to check this, we have: $$\cdots = \partial\overline{\partial}\log(f\overline{f}) = \partial\overline{\partial}\log f + \partial\overline{\partial}\log\overline{f}$$ Since $\log f$ is holomorphic, the first term on the right is 0, so we wish to show that the second term is also zero. Since $\partial,\overline{\partial}$ anti-commute, we get: $$\cdots = -\overline{\partial}\partial\log\overline{f}$$ To show that this is $0$, I'm hoping that $\overline{f}$ being antiholomorphic would imply the same of $\log\overline{f}$ will also be antiholomorphic (which would imply $\partial\log\overline{f} = 0$).
Is this true?
This is one of those situations where the tangent space viewpoint comes in handy even for such seemingly elementary questions.
Here is the tangent space definition of holomorphicity:
Let $X,Y$ be complex manifolds, and $f : X\rightarrow Y$ be a $C^\infty$ map. Thus, at every $x\in X$, from the standard real theory of tangent spaces we get an $\mathbb{R}$-linear map on real tangent spaces $f_*|_x : T_xX\rightarrow T_{f(x)}Y$. Since $X,Y$ are complex manifolds, both tangent spaces have the natural structures of $\mathbb{C}$-vector spaces. We say that $f$ is holomorphic at $x$ if $f_*|_x$ is $\mathbb{C}$-linear. You can find this definition for example in the first chapter of Claire Voisin's excellent book "Hodge theory and complex algebraic geometry I". Similarly, by taking complex conjugates, we say that $f$ is antiholomorphic at $x$ if $f_*|_x$ is $\mathbb{C}$-antilinear (ie, conjugate-linear). One may check that these definitions agree with the usual notion of holomorphicity or anti-holomorphicity defined in terms of charts.
At this point, the desired result follows from the functoriality of tangent spaces. Indeed, given $C^\infty$ maps between complex manifolds $f : X\rightarrow Y$ and $g : Y\rightarrow Z$ and $x\in X$, we find that $(g\circ f)_*|_x = g_*|_{f(x)}\circ f_*|_x$, so if $f,g$ are both either holomorphic or antiholomorphic, then $g\circ f$ is holomorphic if and only if $f,g$ are both holomorphic or $f,g$ are both antiholomorphic. Similarly, $g\circ f$ is antiholomorphic if and only if exactly one of $f,g$ is holomorphic (and the other is antiholomorphic).
At this point, we can check that indeed $\log\overline{f}$ is antiholomorphic by considering the composite $$U\stackrel{\overline{f}}{\longrightarrow}f(U)\stackrel{\log}{\longrightarrow}\mathbb{C}$$