Let $f\colon \mathbb{C} \to \mathbb{C}$ be an entire function taking reals to reals and that is bounded on $\mathbb{R}$. Suppose there exists a continuous function $g\colon \mathbb{R} \to \mathbb{R}$ satisfying $$ |f(z)| \le g(\text{Im}(z)) \quad\quad (*) $$ for every $z\in \mathbb{C}$.
One example is $$ f(z) = e^{-z^2}\,. $$ Of course, if $h\colon \mathbb{C}\to \mathbb{C}$ is an entire function taking reals to reals and satisfying $$ |h(z)|\le k(|z|)\quad\text{for some}~ k\colon\mathbb{R}\to \mathbb{R} ~\text{continuous}\,, $$ and $f$ satisfies ($*$), then the composition $h\circ f$ satisfies ($*$).
Can we say anything else about real entire functions satisfying ($*$)? I'm interested in a Paley–Wiener type theorem for this class of functions but broadly interested in simple examples of this class of entire functions.