Let $f(z) = e^z, g(z) = e^{iz}$ and $S = \{z \in \mathbb C: \mbox{Re }z \in [-\pi, \pi] \}$. Then

Question

Consider the function $f,g:\mathbb C \rightarrow \mathbb C$ defined by $f(z) = e^z, g(z) = e^{iz}$. Let $S = \{z \in \mathbb C: \mbox{Re }z \in [-\pi, \pi] \}$. Then  

1.  $f$ is an onto entire function.
2. $g$ is a bounded function on $\mathbb C$.
3. $f$ is bounded on $S$. 
4. $g$ is bounded on $S$.