Computational Methods and Function Theory 12 (2012), No. 1, 343--361
Copyright Heldermann Verlag 2012
Risto Korhonen
risto.korhonen@uef.fi , University of Eastern Finland, Department of Physics and Mathematics, P.O. Box 111, FI-80101 Joensuu, Finland.
It is shown that if n∈N, c∈Cn, and three distinct values of a meromorphic function f: Cn→P1 of hyper-order σ(f) strictly less than 2/3 have forward invariant pre-images with respect to a translation τ: Cn→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 σ(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→P1 are given, and their applications to partial difference equations are discussed.
Keywords: Picard's Theorem, Second Main Theorem, partial difference equation, several variables, difference analogue.
MSC 2000: Primary 32H25; Secondary 32H30, 39A12, 39B32.
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