What properties function $f$ must have so that $\overline{f(z)} =f(\,\overline {z} \,)$ where $z\in\mathbb{C}$
My working : I can prove it for f being a polynomial in $z$ with real coefficient using conjugates distributing properties. I know how to prove $\overline{\tan z}=\tan \overline z.$
I am searching for a necessary and sufficient condition on $ f$ for which it is true.
Please help me with sufficient but may not be necessary conditions also.
Assuming that $f$ is an entire function, then we have$$(\forall z\in\Bbb C):\overline{f(z)}=f\left(\overline z\right)\tag1$$if and only if $f(\Bbb R)\subset\Bbb R$. In fact, if we $(1)$, then, if $x\in\Bbb R$, $\overline{f(x)}=f\left(\overline x\right)=f(x)$, and so $f(x)\in\Bbb R$. And, if $f(\Bbb R)\subset\Bbb R$, then the function $g\colon\Bbb C\longrightarrow\Bbb C$ defined by $g(z)=\overline{f\left(\overline z\right)}$ is also analytic. But $g(x)=f(x)$ when $x\in\Bbb R$ and therefore, by the identity theorem, $f=g$. But this means that we have $(1)$.