Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 341-361

TROPICAL MEROMORPHIC FUNCTIONS IN A FINITE INTERVAL

Ilpo Laine and Kazuya Tohge

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, FI-80101 Joensuu, Finland; ilpo.laine 'at' joensuu.fi

Kanazawa University, College of Science and Engineering
Kakuma-machi, Kanazawa, 920-1192, Japan; tohge 'at' se.kanazawa-u.ac.jp

Abstract. The tropical Nevanlinna theory in the whole real line R describes value distribution of continuous piecewise linear functions of a real variable with arbitrary real slopes, called tropical meromorphic functions, similarly as value distribution of meromorphic functions of a complex variable is described by the classical Nevanlinna theory in the whole complex plane C. As a tropical counterpart to the Nevanlinna theory in a disc or an annulus centered at the origin, we introduce in this paper a value distribution theory of continuous piecewise linear functions in a symmetric finite open interval (–R,R). The shift operator (difference operator) has a key role in the tropical value distribution theory in R corresponding to the role of the differential operator in the Nevanlinna theory in a subregion of C. However, the affine shift x → x + c does not operate properly in finite intervals. Therefore, we introduce a shift x → s_τ(x) which may be called as the tropical hyperbolic shift. This notion enables us to obtain the quotient estimate m(r,f(s_τ}(x)) \oslash f(x)) = o(1)T(r,f) for tropical meromorphic functions f(x) defined in an interval (–R,R) in R, corresponding to the logarithmic derivative estimate in the Nevanlinna theory for meromorphic functions f(z) defined in a disc or in an annulus. A sort of the second main theorem is also stated by means of this estimate. Concerning hyperbolic shift and the second main theorem, we assume an order restriction to f(x). This restriction is shown to be necessary by an example.

2010 Mathematics Subject Classification: Primary 39A12; Secondary 30D35, 39A13.

Key words: Difference equation, tropical meromorphic function, tropical Nevanlinna theory.

Reference to this article: I. Laine and K. Tohge: Tropical meromorphic functions in a finite interval. Ann. Acad. Sci. Fenn. Math. 44 (2019), 341-361.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4418