Asymptotic Functions of Entire Functions

Aimo Hinkkanen, Joseph Miles, John Rossi

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)619-632
Number of pages14
JournalComputational Methods and Function Theory
Issue number4
StatePublished - Dec 2021


  • Asymptotic function
  • Asymptotic value
  • Entire function

ASJC Scopus subject areas

  • Analysis
  • Computational Theory and Mathematics
  • Applied Mathematics


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