A Difference Picard Theorem for Meromorphic Functions of Several Variables

It is shown that if n ∈ ℕ, c ∈ ℂ n , and three distinct values of a meromorphic function f : ℂ n sr 1 of hyper-order gV( f ) strictly less than 2/3 have forward invariant pre-images with respect to a translation τ: ℂ n sr ℂ n , τ ( z ) = z + c , then f is a periodic function with period c. This resu...

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Bibliographic Details
Published in:Computational methods and function theory Vol. 12; no. 1; pp. 343 - 361
Main Author: Korhonen, Risto
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.06.2012
Subjects:
Analysis
Computational Mathematics and Numerical Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Second Main Theorem
32H30
39B32
difference analogue
Picard’s Theorem
several variables
39A12
32H25
partial difference equation
ISSN:1617-9447, 2195-3724
Online Access:Get full text
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Summary:It is shown that if n ∈ ℕ, c ∈ ℂ n , and three distinct values of a meromorphic function f : ℂ n sr 1 of hyper-order gV( f ) strictly less than 2/3 have forward invariant pre-images with respect to a translation τ: ℂ n sr ℂ n , τ ( z ) = z + c , then f is a periodic function with period c. This result can be seen as a generalization of M. Green’s Picard-Type Theorem in the special case where gV( f ) < 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the Lemma on the Logarithmic Derivative and of the Second Main Theorem of Nevanlinna theory for meromorphic functions ℂ n → ℙ P 1 are given, and their applications to partial difference equations are discussed.
ISSN:1617-9447
2195-3724
DOI:10.1007/BF03321831