criterions for holomorphic functions

You need the chain rule in $\partial/\partial z$ and $\partial/\partial \bar z$ form. If you have $h(z,\bar z) = f(g(z,\bar z))$, then we have $$\frac{\partial h}{\partial \bar z}=\frac{\partial f}{\partial w}\frac{\partial g}{\partial \bar z}+\frac{\partial f}{\partial \bar w}\frac{\partial \bar g}{\partial \bar z}.$$ You can deduce this from the usual chain rule.

By the way, there are identities like $$\overline{\left(\frac{\partial h}{\partial z}\right)}=\frac{\partial \bar h}{\partial \bar z} \quad\text{and} \quad\overline{\left(\frac{\partial h}{\partial \bar z}\right)}=\frac{\partial \bar h}{\partial z}.$$

The extended function

$$g(z) = \begin{cases} f(z) &, \lvert z\rvert \leqslant 1\\ 1/\overline{f(1/\overline{z})} &, \lvert z \geqslant 1\end{cases},$$

where $f \colon \overline{\mathbb{D}} \to \mathbb{C}$ is continuous and holomorphic in $\mathbb{D}$ with $\lvert f(z) \rvert = 1$ for $\lvert z\rvert = 1$, is always a meromorphic function on the entire plane $\mathbb{C}$, with poles in the points obtained from reflection of the zeros of $f$ in the unit circle. That is a special case of the Schwarz reflection principle.

In the particular case that $f$ has no zeros in the unit disk, $f$ is constant by the minimum modulus principle, and the holomorphy (constantness) of $g$ is readily verified.