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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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    $\begingroup$ Andrey Rekalo shows that the answer is all $\theta<\pi$ at mathoverflow.net/questions/29734/… $\endgroup$
    – Jonas Meyer
    Commented Apr 7, 2011 at 22:08
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    $\begingroup$ 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. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 5, 2012 at 9:01

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