TY - JOUR

T1 - Asymptotic Functions of Entire Functions

AU - Hinkkanen, Aimo

AU - Miles, Joseph

AU - Rossi, John

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/12

Y1 - 2021/12

N2 - 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.

AB - 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.

KW - Asymptotic function

KW - Asymptotic value

KW - Entire function

UR - http://www.scopus.com/inward/record.url?scp=85120911284&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85120911284&partnerID=8YFLogxK

U2 - 10.1007/s40315-021-00396-3

DO - 10.1007/s40315-021-00396-3

M3 - Article

AN - SCOPUS:85120911284

VL - 21

SP - 619

EP - 632

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

SN - 1617-9447

IS - 4

ER -