Type: Article
Publication Date: 1969-12-01
Citations: 32
DOI: https://doi.org/10.2140/pjm.1969.31.693
The purpose of this paper is to extend certain theorems on the arithmetic properties of analytic functions due to Straus to functions of several variables.Numerous papers have been written on the arithmetic properties of analytic functions (e.g., Straus [7], Buck [1], Kakeya [3], Selberg [5]).The author is not aware of any analogous studies for analytic functions of several variables.Since the generalization from two to several variables involves no new difficulties that are not already encountered in the generalization from one to two variables, we shall for the sake of simplicity, restrict our discussion to functions of two variables.2* Preliminaries • We begin with a generalization of order and type.DEFINITION 1.Let f(z ly z 2 ) be an entire function of the two variables.Let M(r 19 r 2 ) = M(r) denote the maximum value of |/| on the surface given by | z { \ = r^i -1, 2).(p lf p 2 ) is said to be an order point of /, if for any ε > 0, as r ι + r 2 approaches infinity is bounded, while ikf(r)/exp (rfi + r£ 2 ~ε) and Λf(r)/exp (r^~ε + rξή are both unbounded.The set, p, of all such points (p ίf ρ 2 ) is called the order of /.DEFINITION 2. Let f(z 19 z 2 ) be as above and let (ρ 19 p 2 ) be one of its order points.(σ 19 σ 2 ) is said to be a type point of / at (p t1 p 2 ) if for any ε > 0, as r 1 + r 2 approaches infinity