Asymptotic functions of entire functions

Let $f(z)$ be an entire function in the complex plane ${\mathbb C}$. An entire function $a(z)$ is said to be an asymptotic function for $f$ if there exists a path $\gamma$ in ${\mathbb C}$ from $0$ to infinity such that $f(z)-a(z)$ tends to $0$ as $z$ tends to infinity along $\gamma$. If $a(z)$ is a constant function, the value $a$ is said to be an asymptotic value of $f$. The Denjoy–Carleman–Ahlfors Theorem states that if $f$ has $n$ distinct asymptotic values then the rate of growth of $f$ is at least order $n/2$, mean type. This bound is known to be sharp. For asymptotic functions, the best general known result guaranteeing the same conclusion regarding $f$ is that it suffices to assume that the rate of growth of each $a(z)$ is at most order $1/4$, minimal type; order $1/2$ minimal type would be best possible but this remains open. For special configurations of paths, somewhat stronger results than order $1/4$ have been obtained by various authors.
We obtained some conditions on the function $f$ and associated asymptotic paths that are sufficient to guarantee that $f$ satisfies the conclusion of the Denjoy–Carleman–Ahlfors Theorem for asymptotic functions of suitable growth. In addition, we proved that for each positive integer $n$, and for any $n$ distinct, prescribed asymptotic functions of order less than $1/2$, there exists an entire function of order $n$ having these asymptotic functions.
This is joint work with Joseph Miles and John Rossi.