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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 $0<p<1$? 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.

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I added the computational-complexity tag since that's such a locus of interest in questions like these. – Kevin Carlson Nov 1 '12 at 8:24
up vote 7 down vote accepted

One of the best-known classes is the "quasi-polynomials", which are exponentials of polynomials in logs, e.g. $e^{\log^2(x)+\log x}$, which you might also write as $x^{\log(x)+1}$. As long as the degree of the exponent is greater than $1$, these fit between polynomial and exponential.

One has also the "sub-exponentials," which grow as $e^\phi$ where $\lim\limits_{x\to \infty}\frac{\phi(x)}{x}=0$. The most obvious examples that aren't quasi-polynomial are along the lines of the one you gave.

These don't exhaust the possibilities, though. You may be interested in a considerable volume of discussion over at MO of functions $f$ such that $f(f(x))$ is exponential.

These "half-exponentials" are in between the two classes I've described: a proper sub-exponential has an exponent that dominates all polynomials of logarithms, so its composition with itself has an exponent that dominates all polynomials-and thus isn't exponential. In the other direction, you can see that quasi-polynomials are closed under self-composition. Here's the latest thread, with links to others.

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Where you write "One has also the 'sup-exponentials'"... do you mean sub-exponential? I ask because you use the latter term later. – Alex Reinking Jun 5 at 21:32
I think I have two typos here. – Kevin Carlson Jun 6 at 1:18

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