If $f(z)$ is analytic, and $\overline{f(z)}$ is analytic, then is $f$ necessarily a constant function

Question

I need help with verifying my following proof. It feels a little fishy to me.

If $f(z)$ is analytic, and $\overline{f(z)}$ is analytic, then is $f$ necessarily a constant function?

We know $f(z)=u(x,y)+iv(x,y)$ and $\overline{f(z)}=u(x,y)+iv'(x,y)$, where $v'=-v$. $f$ satisifies the Cauchy Riemann equations, thus,

For $f$, one has that: $u_x=v_y, v_x=-u_y$.

For $\overline{f}$, one has that:

$u_x=v'_y=-v_y$

$v'_x=-v_x=–u_y$.

One has $u_x=-v_y=v_y$, which forcefully makes $v_y=0$. Also, $u_y=v_x=-v_x$, so $v_x=0$. So for all $z$, $f'(z)=0$ and this shows that $f$ is a constant function.

Does this proof work?

Answer 1

Your proof is correct. My proof: let $g(z):=f(z)\overline{f(z)}.$ Then $g(z)=|f(z)|^2$ is analytic and real(!). Conclusion ?