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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: [email protected]
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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

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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