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Asymptotic Functions of Entire Functions

Asymptotic Functions of Entire Functions 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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma $$\end{document} from 0 to infinity such that f(z)-a\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(z) - a$$\end{document} tends to 0 as z tends to infinity along γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma $$\end{document}. 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Asymptotic Functions of Entire Functions

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-021-00396-3
Publisher site
See Article on Publisher Site

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 γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma $$\end{document} from 0 to infinity such that f(z)-a\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(z) - a$$\end{document} tends to 0 as z tends to infinity along γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma $$\end{document}. 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.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Dec 1, 2021

Keywords: Entire function; Asymptotic function; Asymptotic value; Primary 30D20; Secondary 31A05

References

  • Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung
    Ahlfors, LV
  • Über die asymptotischen Werte der meromorphen Funktionen endlicher Ordnung
    Ahlfors, LV
  • Entire functions having asymptotic functions
    Fenton, PC
  • Phragmén-Lindelöf theorems
    Fenton, PC; Rossi, J
  • A reverse Denjoy theorem
    Fenton, PC; Rossi, J
  • A reverse Denjoy theorem II
    Fenton, PC; Rossi, J
  • A reverse Denjoy theorem III
    Fenton, PC; Rossi, J
  • A reverse cosπρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos \pi \rho $$\end{document} theorem and a question of Fryntov
    Fenton, PC; Rossi, J
  • Entire functions with asymptotic functions
    Hinkkanen, A; Rossi, J