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**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.

(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 |

### Keywords:

maximal differential subfields; algebraic differential equation; Hardy’s L-functions; Orders of infinity; growth properties### Citations:

Zbl 0539.26003
Full Text:
DOI

### References:

[1] | Barness, E. W., The theory of the Gamma-function, Messenger of Mathematics, XXIX, 122-128 (1900) |

[2] | Cohn, P. M., Algebra, Vol. 2 (1977), London: Wiley, London · Zbl 0341.00002 |

[3] | Hardy, G. H., Properties of logarithmico-exponential functions, Proc. London Math. Soc., 10, 2, 54-90 (1912) · JFM 42.0437.02 |

[4] | Hardy, G. H., Some results concerning the behavior at infinity of a real and continuous solution of an algebraic differential equation of the first order, Proc. London Math. Soc., 10, 2, 451-468 (1912) · JFM 43.0390.02 |

[5] | G. H. Hardy,Orders of Infinity, Cambridge Tracts in Math. and Math. Phys. 12 (2nd edition), Cambridge, 1924. · JFM 50.0153.04 |

[6] | Szekeres, G., Fractional iteration of exponentially growing functions, J. Austral. Math. Soc., 2, 301-320 (1961) · Zbl 0104.09601 |

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