Asymptotic functions of entire functions

TL;DR

This paper investigates the growth behavior of entire functions with multiple asymptotic values or functions, extending classical theorems and providing new conditions and examples related to their asymptotic properties.

Contribution

It establishes new sufficient conditions for entire functions to satisfy the Denjoy–Carleman–Ahlfors theorem and constructs examples with prescribed asymptotic functions of lower order.

Findings

Conditions on functions and paths ensure the theorem's conclusion

Constructed examples of entire functions with prescribed asymptotic functions

Extended understanding of asymptotic behavior in entire functions

Abstract

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 $1/2$.