# Entire function bounded in the upper half-plane

Let $f(z)$ be an entire function which is real on the real axis, with $f(0) = 1$. Assume that there is a real constant $M$ such that $|f(z)| \le M$ for all $z$ satisfying $Im(z) \ge 0$. Evaluate $$\lim\limits_{x \rightarrow + \infty}{f(x)}.$$

It's obviously about Liouville's theorem, but we know nothing about $f(z)$ in the other half-plane. I tried to consider $g(z) = f(\sqrt{z})$ ($\sqrt{z}$ is a principal branch), which is bounded everywhere, but $\sqrt{z}$ is not entire (it is not even defined on $[0; +\infty)\ )$ and therefore $g(z)$ is not entire. I would also be very grateful if someone told me some properties of real entire functions - I found almost nothing about them.

-

but we know nothing about $f(z)$ in the other half-plane.

We do:

Let $f(z)$ be an entire function which is real on the real axis

That means we have

$$f(z) = \overline{f(\overline{z})}$$

by the identity theorem, and hence a global bound.

-