Question

Let ${z_n}$ be an infinite sequence of complex numbers. Under which conditions on these numbers does there exist an entire function $f$ such that the $z_n$ are the zeros of $f$ and $|f(z)|< C \exp(c|\Im z|)$ for some constants $C,c>0$?

Answer 1

It is unlikely that a simple explicit necessary and sufficient condition exists. But some complicated condition is given in the paper MR2411971 Favorov, S. Yu. Zero sets of entire functions of exponential type with additional conditions on the real line. Algebra i Analiz 20 (2008), no. 1, 138--145 (Russian). Translation in St. Petersburg Math. J. 20 (2009), no. 1, 95–100.

This condition is necessary and sufficient but probably hard to verify for any specific set.

The class of entire functions you are interested in is called Bernstein's class B in this paper.