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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 \begin{equation} \lim\limits_{x \rightarrow + \infty}{f(x)}. \end{equation}

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.

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up vote 4 down vote accepted

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.

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