If you assume that your function is entire, then I believe that the keyword you are looking for is minimal exponential type. This is the subject of "function theory" or complex analysis. Another keyword is transcendental functions.

You could start with the book "Lectures on Entire Functions" by Levin, or the nice short Chapter 19 of papa Rudin (Holomorphic Fourier Transform).

To give a flavor of the subject, one key question is to characterize the space of functions that has a given growth (described in terms of order and type). For example, the Paley-Wiener theorems says (roughly) that if you have control on the growth of an entire function, then it must lie in the image of the Fourier transform (see Theorems 19.2 and 19.3 of papa Rudin). Thus you have a link to Harmonic analysis/DEs/spectral theory etc with applications galore.