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.