  If $f$ is an entire function and $a$ is a complex number, $a$ is said to be
an asymptotic value of $f$ if there exists a path $\gamma$ from $0$ to infinity
such that $f(z) - a$ tends to $0$ as $z$ tends to infinity along $\gamma$. The
Denjoy--Carleman--Ahlfors Theorem asserts that if $f$ has $n$ distinct
asymptotic values, then the rate of growth of $f$ is at least order $n/2$, mean
type. A long-standing problem asks whether this conclusion holds for entire
functions having $n$ distinct asymptotic (entire) functions, each of growth at
most order $1/2$, minimal type. In this paper conditions on the function $f$
and associated asymptotic paths are obtained that are sufficient to guarantee
that $f$ satisfies the conclusion of the Denjoy--Carleman--Ahlfors Theorem. In
addition, for each positive integer $n$, an example is given of an entire
function of order $n$ having $n$ distinct, prescribed asymptotic functions,
each of order less than $1/2$.
