Suppose $f(z)$ is holomorphic on $\Re(z) \geq 0$ satisfying $f(-it) = -f(it)$ for any $t \in \mathbb{R}$. Can $f$ be analytically continued to an entire function $g$ on $\mathbb{C}$ satisfying $g(-z) = -g(z)$ for any $z \in \mathbb{C}$?
I believe this should hold with $$ g(z) = \begin{cases} f(z) &\mbox{ if } \Re(z) \geq 0\\ -f(-z)&\mbox{ otherwise. } \end{cases} $$ It is clearly holomorphic on both $\Re(z) > 0$ and $\Re(z) < 0$. It remains to show it is holomorphic on a neighbourhood containing $i\mathbb{R}$, which should follow from the constraint above. But I'm not sure how to make this precise.