Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T03:35:54.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Gaussian integer points of analytic functions in a half-plane

Published online by Cambridge University Press:  01 September 2008

ALASTAIR FLETCHER
ALASTAIR FLETCHER*
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD. e-mail: alastair.fletcher@nottingham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

A classical result of Pólya states that 2z is the slowest growing transcendental entire function taking integer values on the non-negative integers. Langley generalised this result to show that 2z is the slowest growing transcendental function in the closed right half-plane Ω = {z ∈ : Re(z) ≥ 0} taking integer values on the non-negative integers. Let E be a subset of the Gaussian integers in the open right half-plane with positive lower density and let f be an analytic function in Ω taking values in the Gaussian integers on E. Then in this paper we prove that if f does not grow too rapidly, then f must be a polynomial. More precisely, there exists L > 0 such that if either the order of growth of f is less than 2 or the order of growth is 2 and the type is less than L, then f is a polynomial.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 2 , September 2008 , pp. 257 - 272
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bank, S. B. and Langley, J. K.. On the value distribution theory of elliptic functions. Monatsh. Math. 98 no. 1 (1984), 1–20.CrossRefGoogle Scholar
[2]Beardon, A. F. and Minda, D.. The hyperbolic metric and geometric function theory. Quasiconformal Mappings and their Applications (Narosa Publishing House, 2007).Google Scholar
[3]Boas, R. P. Jr.Entire Functions (Academic Press Inc., 1954).Google Scholar
[4]Buck, R. C.. Integral valued entire functions. Duke Math. J. 15 (1948), 879–891.CrossRefGoogle Scholar
[5]Fletcher, A. and Langley, J.. Integer points of analytic functions in a half-plane. Proc. Edin. Math. Soc., to appper.Google Scholar
[6]Gelfond, A. O.. Transcendental and Algebraic Numbers (Dover Publication, 1960).Google Scholar
[7]Gol'dberg, A. A. and Ostrovskii, I. V.. Distribution of values of meromorphic functions (Nauka, 1970).Google Scholar
[8]Langley, J. K.. Integer points of meromorphic functions. Comput. Methods Funct. Theory 5 (2005), 253–262.CrossRefGoogle Scholar
[9]Langley, J. K.. Integer points of entire functions. Bullet. London Math. Soc. 38 (2006), 239–249.CrossRefGoogle Scholar
[10]Langley, J. K.. Integer-valued analytic functions in a half-plane. Comput. Methods Funct. Theory 7 (2007), 433–442.CrossRefGoogle Scholar
[11]Pólya, G.. Über ganze ganzwertige Funktionen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1921), 1–10.Google Scholar
[12]Robinson, R. P.. Integer-valued entire functions. Trans. Amer. Math. Soc. 153 (1971), 451–468.CrossRefGoogle Scholar
[13]Waldschmidt, M.. Integer valued entire functions on Cartesian products. Number Theory in Progress, Vol. 1 (Zakopane-Koscielisko 1997), 553–576 (de Gruyter, 1999).CrossRefGoogle Scholar
[14]Welter, M.. A new class of integer-valued entire functions. J. Reine Angew. Math. 583 (2005), 175–192.CrossRefGoogle Scholar
[15]Whittaker, J. M.. Interpolatory Function Theory (Cambridge Tract No. 33, Cambridge University Press, 1935).Google Scholar