Abstract
It is shown that if n ∈ ℕ, c ∈ ℂn, and three distinct values of a meromorphic function f: ℂn sr 1 of hyper-order gV(f) strictly less than 2/3 have forward invariant pre-images with respect to a translation τ: ℂn sr ℂn, 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 ℂn → ℙ P1 are given, and their applications to partial difference equations are discussed.
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The research reported in this paper was supported in part by the Academy of Finland grant #118314 and #210245, the Isaac Newton Institute for Mathematical Sciences, and the Nord-Forsk foundation.
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Korhonen, R. A Difference Picard Theorem for Meromorphic Functions of Several Variables. Comput. Methods Funct. Theory 12, 343–361 (2012). https://doi.org/10.1007/BF03321831
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DOI: https://doi.org/10.1007/BF03321831