Nevanlinna theory for the difference operator
By R. Halburd, R. Korhonen. 2005
447 citations recorded by the source corpus.
Certain estimates involving the derivative f 7→ fof a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to extend the usual Nevan- linna theory to a theory for the exact difference f 7→ �f = f(z + c) − f(z). An a-point of a meromorphic function f is said to be c-paired at z ∈ C if f(z) = a = f(z+c) for a fixed constant c ∈ C. In this paper the distribution of paired points of finite-order meromorphic functions is studied. One of the main results is an analogue of the second main theorem of Nevanlinna the- ory, where the usual ramification term is replaced by a quantity expressed in terms of the number of paired points of f. Corollaries of the theorem include analogues of the Nevanlinna defect relation, Picard's theorem and Nevanlinna's five value theorem. Applications to difference equations are discussed, and a number of examples illustrating the use and sharpness of the results are given.
Subfields: Applied Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/ad6761c9dcd8836439eb084703cc204bc2bfa657
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In Applied Mathematics
In Numerical Analysis
Citations in the corpus
- Value distribution of the difference operator (2009, background): “Halburd and Korhonen [8] established a version of Nevanlinna theory based on difference operators.”
- Finite‐order meromorphic solutions and the discrete Painlevé equations (2005, result): “The results described in [18] have led to difference analogues of a number of other important theorems from Nevanlinna theory, including the Second Main Theorem [19].”
- Holomorphic curves with shift-invariant hyperplane preimages (2009, background): “The difference has caused different choices of an auxiliary function which is essential in the two papers, which might be observed by the expressions L in (2.3) and L̃ in (9.3) here in this paper.”
- Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity (2009, methodology): “We proceed to find alternative improvements of Theorem B by means of Nevanlinna theory for exact differences [9].”
- A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables (2016, background): “Korhonen [4] considered the second main theorem for complex difference operator with finite order in the complex plane.”
- Zeros of differences of meromorphic functions (2005, background): “Applications of these estimates appear in [5, 10, 11].”
- Value sharing results for shifts of meromorphic functions (2011, background): “Currently, J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. Zhang [8] obtained that if f (z) is of finite order and shares two values CM and one value IM (Ignoring multiplicities) with its shift f (z + c), then f is a periodic function with period c.”
- Uniqueness of meromorphic functions sharing values with their shifts (2011, background): “3 [17] Let f be a meromorphic function of finite order, c2C and let 5 1.”