Let $f : \mathbb C^n \rightarrow \mathbb C^m$ be holomorphic and $g : \mathbb C^m \rightarrow \mathbb C$ be smooth. I am looking for a simple formula for the mixed partials $\partial_i \partial_{\bar j}(g\circ f)$, where $\partial_i$ and $\partial_{\bar j}$ denote complex differentials,
$\partial_i u = \frac{\partial u}{\partial x_i} - \sqrt{-1} \frac{\partial u}{\partial y_i}$
and
$\partial_{\bar j} u = \frac{\partial u}{\partial x_j} + \sqrt{-1} \frac{\partial u}{\partial y_j}.$
Applying the chain rule seems to give me a rather nasty looking expression involving summation over several different indices. Is there a simpler way to write this?