If $f$ and $1/f$ are harmonic then $f$ is holomorphic or antiholomorphic I have this problem.
Let $f:D\to \mathbb{C}$ be a function such that $f$ and $1/f$ are harmonic (Their real and imaginary parts are harmonic). Then $f$ is holomorphic or antiholomorphic.
I tried to solve it by computing the laplacian of real and imaginary parts, but it becomes very cumbersome. Is there a better way?

Let $\frac{\mathrm{d}}{\mathrm{d}z}=\tfrac12(\frac{\mathrm{d}}{\mathrm{d}x}-\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}y})$ and $\frac{\mathrm{d}}{\mathrm{d}\overline{z}}=\tfrac12(\frac{\mathrm{d}}{\mathrm{d}x}+\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}y})$ as usual. Then $$\frac{\mathrm{d}^2}{\mathrm{d}z \, \mathrm{d}\overline{z}} = \frac14\left(\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ \frac{\mathrm{d}^2}{\mathrm{d}y^2}\right)$$ and $f$ is holomorphic if $\frac{\mathrm{d}f}{\mathrm{d}\overline{z}}=0$ or anti-holomorphic if $\frac{\mathrm{d}f}{\mathrm{d}z}=0$. Now under the given conditions $$0 = \frac{\mathrm{d}^2 f^{-1}}{\mathrm{d}z \, \mathrm{d}\overline{z}} = 2 f^{-3} \frac{\mathrm{d}f}{\mathrm{d}z} \frac{\mathrm{d}f}{\mathrm{d}\overline{z}}.$$
