Show that a smooth complex function $f(z, \bar{z})$ satisfies $\frac{\partial^2 f}{\partial \bar z^{2}}=0$ if and only if $f(z, \bar{z})=\bar{z}g(z)+h(z)$ for some analytic functions $g(z)$ and $h(z)$.
Thought: $(\Leftarrow):$ Follows from taking derivative with respect to $\bar{z}$, since $\frac{\partial f}{\partial \bar z}=g(z)+\bar{z} \frac{ \partial g}{ \partial \bar{z}}+\frac{\partial h}{\partial \bar z}$ and $\frac{\partial^2 f}{\partial \bar z^{2}}=\frac{\partial g}{\partial \bar z}=0$, where I used the fact that $g $ and $h$ are both analytic functions, i.e, $\frac{\partial g}{\partial \bar z}=0$ and $\frac{\partial h}{\partial \bar z}=0$. I have trouble in the other direction. Thanks in advance.
Assume $$ \frac{\partial^2 f}{\partial \overline z^2} = 0.\qquad(\star) $$ Set $$ g(z):=\frac{\partial f}{\partial\overline z}. $$ You know from the assumption $(\star)$ that $g(z)$ is holomorphic (i.e., analytic). Define $$ h(z):=f(z)-\overline z g(z). $$ Compute $$ \frac{\partial h}{\partial \overline z}=... $$