What is known in contemporary mathematics about orders of growth for functions that exceed any degree polynomial, but fall short of exponential? This is a subject for which I've found little literature in the past.
An example: $Ae^{a\sqrt x}$ clearly will outrun any finite degree polynomial, but will be outrun by $Be^{bx}$.
If we replace $x$ with $y^2$ then that example doesn't seem so deep. Are there functions that exceed polynomial growth yet fall short of $Ae^{ax^p}$ for any power $0 ? What classes of functions can we distinguish with different kinds of in-between orders of growth? What can we know about their power series expansions, or behavior in the complex plane? Those are examples of the kinds of questions I have, and would like to find literature on. Have any definitions or terminology been established concerning this? The right jargon will facilitate searching.