Tropical meromorphic functions in a finite interval

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) ⊘ 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.