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Define $W : [0,\infty) \to 2^{\mathbb{C}}$ by $W(\theta) = \{r\cdot \operatorname{exp}(i\cdot t) : \langle r,t \rangle \in (0,\infty) \times (-\theta,\theta)\}$.
For what $\theta$ does there exist an entire function $f$ such that $\{f(z) : z\in W(\theta)\}$ is bounded?

I know this holds for $\theta \leq \frac12 \cdot \pi$, because of $(z\mapsto \operatorname{exp}(-z))$,
and fails for $\pi \leq \theta$, because $W(\theta)$ would be dense.

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closed as no longer relevant by Ricky Demer, S. Carnahan Aug 6 '12 at 1:07

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Andrey Rekalo shows that the answer is all $\theta<\pi$ at… – Jonas Meyer Apr 7 '11 at 22:08
Examples of such functions are Mittag-Leffler functions, and they are described in good textbooks on function theory or on special functions. I propose to close the question. – Alexandre Eremenko Aug 5 '12 at 9:01