Half-exponential function

In mathematics, a half-exponential function is a functional square root of an exponential function, that is, a function "ƒ" that, if composed with itself, results in an exponential function:
Another definition is that "ƒ" is half-exponential if it is non-decreasing and "ƒ"−1("x""C") ≤ o(log "x"). 
for every "C" > 0.
It has been proven that if a function "ƒ" is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then "ƒ"("ƒ"("x")) is either subexponential or superexponential. Thus, a Hardy "L"-function cannot be half-exponential.
There are infinitely many functions whose self-composition is the same exponential function as each other. In particular, for every formula_2 in the open interval formula_3 and for every continuous strictly increasing function "g" from formula_4 onto formula_5, there is an extension of this function to a continuous strictly increasing function formula_6 on the real numbers such that formula_7. The function formula_6 is the unique solution to the functional equation
A simple example, which leads to "ƒ" having a continuous first derivative everywhere, is to take formula_10 and formula_11, giving 
Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.