2021/02/22 by Aimo Hinkkanen, Hinkkanen, Aimo, Joseph Miles +3
Mathematics · #30D20 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30D20
paper · pdf · doi:10.48550/arxiv.2102.11332
arxiv created 2021/02/22 · arxiv updated 2021/02/24
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 γ from 0 to infinity such that f(z) - a tends to 0 as z tends to infinity along γ. 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.