Hardy fields and existence of transexponential functions
The main result of this paper is that every Hardy field, which is maximal [cf. \textit{N. Boubaki}: Eléments de mathématique. Fasc. XII. Première partie. Livre IV: Fonctions d'une variable réelle (théorie élémentaire) (1961; Zbl 0131.050), p. 107] under \(\gg\) (greater for sufficiently large values of the variable), is unbounded. It is also shown that there exist transexponential functions \((f\gg e_ k\) for all positive k, where \(e_ k\) is the k-th iterate of the exponential function) in Hardy fields and that there exist (transexponential) analytic solutions f with \(\lim f=\infty,\) \(f\gg 0,\) of the functional equation \(f(x+1)=e^{f(x)}\) and also of \(f(x+1)=e^{f(x)}-1,\) such that solutions of the former equation satisfy \(f(x)f'(x-1)=f'(x)\) (the statements in the paper are stronger). Finally properties of rational iterates of the exponential function are examined and four unsolved problems are listed (a note added in proof states that the author has since solved the first two).
rational iterates of the exponential function
scientific article; zbMATH DE number 3953127
scientific article