Existence of a non-constant entire function

a. $e^{iz}$ satisfies $|e^{iz}|=e^{-\text{Im}(y)}<1$

b. $iz$ as Tim Raczkowski told you.

c. $e^{z}$ is bounded on the imaginary axis. If $iz\in\mathbb{R}$ then $|e^{z}|=1$.

Part b) is incorrect. $f(z)=iz$ takes on real values on the imaginary axis.