Denjoy–Carleman–Ahlfors theorem

Remarks

Between two curves along which $f$ tends to different finite limits, $f$ must be unbounded, so $f$ has at least $n$ separate regions near $\infty$ where $\abs{f}$ is large. Ahlfors's proof bounds the growth of $f$ in such regions with an estimate for conformal maps of strip-like domains. This estimate is now called the Ahlfors distortion theorem. It is proved by the length–area method, which later grew into the theory of extremal length. For example, $e^z$ has order $1$ and the single finite asymptotic value $0$, reached along the negative real axis, so $1 \le 2$. For $n = 1$ the theorem says that an entire function of order less than $1/2$ has no finite asymptotic value.

Going deeper

For a student

The order is defined by $\rho = \limsup_{r \to \infty} \log \log M(r)/\log r$, where $M(r)$ is the maximum of $\abs{f}$ on $\abs{z} = r$. The case $n = 1$ is consistent with the $\cos \pi\rho$ theorem, which shows that functions of order less than $1/2$ are large on a sequence of whole circles. The method of the proof, through extremal length and conformal invariants, became a central tool of geometric function theory.