Let $f(z)$ be an analytical function on the whole complex plane, with $\Im f(z) = 0$ when $\Im z = 0$ and $\Im f(z) \ge 0$ when $\Im z > 0$. It can be found quickly that linear functions $f(z)=az + b$, where $a \ge 0, b \in \mathbb{R}$ satisfy the requirements. Are there any other possibilities? Can we find a general form of $f(z)$?
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Well, $e^{if}$ is then bounded in the upper half plane. Don't know where that leads - just throwing it out there. – JonathanZ Jun 07 '22 at 14:42
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Check this: https://math.stackexchange.com/q/2634606/42969 – Martin R Jun 07 '22 at 15:24
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@MartinR thanks! That's what I want – Rintarou Jun 07 '22 at 16:52