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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Vydané v:	Computational methods and function theory Ročník 12; číslo 1; s. 343 - 361
Hlavný autor:	Korhonen, Risto
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer-Verlag 01.06.2012
Predmet:	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
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Shrnutí:	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