If $D$ is a domain symmetric about the real-axis and $f:D \rightarrow \mathbb{C}$ analytic, then $\overline{f(\overline{z})}$ is analytic

Question

My original problem that I encounter was:

"Let $D$ be a domain symmetric about the real-axis and containing the real axis. Suppose that $f:D \rightarrow \mathbb{C}$ analytic on $D$ and that when $x$ is real, $f(x)$ is pure imaginary. Show $-\overline{f(z)}=f(\overline{z})$."

Now, my idea has been to show that $\overline{f(\overline{z})}$ and $-f(z)$ are both analytic on $D$. Then, since these two are equal on the real-axis, the must be equal everywhere. However, I'm having trouble showing that $f$ analytic implies that $\overline{f(\overline{z})}$ is also analytic. I'm not sure how $f$ being able to be expanded as a power series around points in $D$ means that $\overline{f(\overline{z})}$ can be expanded as a power series around points in $D$. Could anyone let me know how I should proceed from here?

Would this help? How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is? — Bruno B, Sep 20 '23 at 19:32
If $f(z)$ can be expanded locally as a power series $\sum_{n=0}^\infty a_n(z-z_0)^n$ on some open disc $B(z_0,r)\subseteq D$, you need to prove that $\overline{f(\overline{z})}$ can also be expanded locally as a power series about $z_0$. — user1551, Sep 20 '23 at 20:07
@BrunoB, that was incredibly helpful, thank you! I completely forgot that conjugation is a continuous function so the limit can be pushed inside the summation! — obitobi_tobias, Sep 20 '23 at 22:30

If $D$ is a domain symmetric about the real-axis and $f:D \rightarrow \mathbb{C}$ analytic, then $\overline{f(\overline{z})}$ is analytic

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