We consider real-valued functions ƒ(x) which are defined for all sufficiently large real numbers x. In discussing the behaviour of such functions as x → + ∞, it is useful to compare ƒ with the functions of some “comparison scale”. The early work in this field was due to Du Bois-Reymond (see, for example, (2), (3)). This was elaborated by Hardy, (6), who was mainly concerned with what he calls the “logarithmico-exponential” scale of functions. This “scale of Hardy” may be defined as the smallest class ℋ of functions ƒ with the following properties:
(i) if ƒ ∈ ℋ, ƒ, is defined and continuous for all sufficiently large values of x;
(ii) (a) the function ƒ(x) ≡ α, where α is any real constant, is in ℋ;
(b) the function ƒ(x) ≡ x is in ℋ;
(iii) if ƒ and g are in ℋ and g is non-zero for all sufficiently large values of x, then
are also in ℋ.