TITLE: Asymptotic inverses of asymptotic functions
QUESTION [5 upvotes]: Under what conditions are the inverses of asymptotic functions themselves asymptotic? (A simple example where this is not the case is the pair of functions $\log (x)$ and $\log (2x)$.)

REPLY [2 votes]: In fact, it's a problem about asymptotic solution of transcendental equations.
For equations with real variables, we have the following conclusion.

$\mathbf{Theorem\ 1.}$ Let $f(\xi)$ be continuous and strictly increasing in an interval $a<\xi<\infty$, and 
  $$f(\xi)\sim \xi, \ \xi \to \infty.$$
  Denote by $\xi(u)$ the root of the equation $f(\xi)=u$ which lies in $(a,\infty)$ when $u>f(a)$.Then
  $$\xi(u)\sim u,\ u\to \infty.$$

It's fortunate to generalize this conclusion to complex variables, but there're several constraints.

$\mathbf{Theorem\ 2.}$ Suppose that $f(z)$ is an analytic function of complex $z$, which is holomorphic in a region containing a closed annular sector $\mathbf{S}$ with vertex at the origin and angle less that $2\pi$. 
  $$\ $$
  Let $\mathbf{S}_1$ and $\mathbf{S}_2$ be closed annular sectors with vertices at the origin, $\mathbf{S}_1$ being properly interior to $\mathbf{S}$, and $\mathbf{S}_2$ being proper interior to $\mathbf{S}_1$.Then
  $$z(u)\sim u, as\ u\to \infty\ in\ \mathbf{S}_2,$$
  where $z(u)$ is a root of the equation $u=f(z)$.