Suppose a complex valued function $f$ is entire, maps $\mathbb{R}$ to $\mathbb{R}$, and maps the imaginary axis into the imaginary axis.
I see that $f(x)=\overline{f(\bar{x})}$ on the whole real axis, and thus the identity theorem implies that $f(z)=\overline{f(\bar{z})}$ for all $z\in\mathbb{C}$. Then for $ai$ on the imaginary axis, it follows that $ f(ai)=\overline{f(\overline{ai})}=\overline{f(-ai)}=-f(-ai). $
Does this relation somehow extend to arbitrary $z$ so that $f$ is odd on the whole complex plane?