## An extension of Hardy’s class L of ”orders of infinity”.(English)Zbl 0539.26002

Let B denote the ring of germs of continuous real-valued functions at infinity. There is a natural partial ordering on the ring. Two functions $$f,g\in B$$ are called comparable if the difference (f-g) is ultimately $$(=$$ for all x sufficiently large) of definite sign and $$f\gg g$$ if it is ultimately positive. This ordering plays an important role in the author’s considerations. Particularly, every subfied of B containing real constants only (only such fields are considered) is an ordered field with respect to the relation $$\gg$$. Let $$C^ SB$$, $$C^{\infty}B$$ and $$C^ AB$$ denote the differential subrings of B consisting of the sufficiently smooth (upper index means smoothness degree) functions at infinity. The intersection of all maximal differential subfields of such CB ring forms a differential field $$E=E(CB)$$. The field E has a number of significant properties. For example, the author proved (Theorems 6.1; 6.8; 10.1) that:
(1) E is closed under composition of functions;
(2) each function $$f\in E$$ is ultimately (real) analytic, satisfies an algebraic differential equation and moreover, there exists an iterate $$\exp(\exp(...\exp(x)...))=g(x)$$ of $$e^ x$$, such that $$g\gg f$$;
(3) E is a real closed field;
(4) if $$f\in E$$, then $$e^ f\in E$$;
(5) if $$f\in E$$ and $$f\neq 0$$, then $$\ln | f| \in E$$;
(6) if $$f\in E,$$ then $$\int^{x}f(t)dt\in E;$$
(7) $$L\subset E$$, where L denotes the class of Hardy’s L-functions [see G. H. Hardy, ”Orders of infinity” (Cambridge, 1924)]. The consideration of properties of E takes a central place in the paper and are applied to study the growth properties of real solutions of the differential equation $$y'=h(x,y)$$ at infinity. For this equation the author obtains the results which are stronger and more general than the corresponding results of Hardy [see G. H. Hardy, Proc. Lond. Math. Soc., II. Ser. 10, 451-468 (1912)]. In conclusion the author formulated following conjectures: $$\quad E(C^ SB)=E(C^{\infty}B)=E(C^ AB);$$ 2) E contains a differential closure of $${\mathbb{R}}$$ in CB; 3) E is closed under compositional inversion.
Reviewer: N.V.Grigorenko

### MSC:

 26A12 Rate of growth of functions, orders of infinity, slowly varying functions 12H05 Differential algebra 34C11 Growth and boundedness of solutions to ordinary differential equations

Zbl 0539.26003
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### References:

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