In the previously published theory of higher-order types of asymptotic variation, [1]-[4], we started from the elementary theory of regular and rapid variation for those restricted classes of functions absolutely continuous on some interval [ T , + ∞ ) and such that the “ lim x → + ∞ x f ′ ( x ) / f ( x ) ” exists as an extended real number α labelled as the “index of variation” at + ∞ . Higher-order regular or rapid variation was then defined by assuming similar asymptotic conditions for some of the derivatives, having preliminarily established precise relationships between the indices of the involved derivatives. A similar procedure was adopted for rapid variation and for three types of exponential variation. This is an approach going back to Hardy [5] for the first order and sketched in Bourbaki ([6], Prop. 2, p. V.40) for higher orders, but in a different context.

The standard Karamata theory of regular (and, for extension, rapid) variation involves the larger classes of functions assumed Lebesgue-measurable on [ T , + ∞ ) and such that “ lim x → + ∞ f ( λ x ) / f ( x ) ” exists and equals λ α for all λ > 0 , including the limit cases “ α = ± ∞ ” for rapid variation. Other approaches will be mentioned below. In all of these frameworks, all the functions are traditionally assumed positive in a neighborhood of + ∞ .

In the case of regular variation, Hardy’s approach is based on the “asymptotic differential equation”.

whereas Karamata’s approach starts from the “asymptotic functional equation”.

Solutions of (1.1) form a subclass of the solutions of (1.2), and the class of solutions of (1.2) happens to be the “closure” of the solutions of (1.1) with respect to the equivalence relation of “same growth-order” in the classical sense of

see Propositions 2.1 and 2.3 below.

Something analogous happens for the various concepts of rapid and exponential variation, and the reader of the higher-order theory might ask why Equation (1.2) referred to f ′ was not used instead of (1.1), with f replaced by f ′ , in constructing the advanced theory. And similar remarks might apply to the other approaches. In this paper, we:

(I) first point out the relationships between the known approaches to the concept of regular variation;

(II) and then show that a higher-order theory of regular variation according to any of these approaches is either of little import or equivalent to the theory developed by the author according to Hardy’s approach.

Answering to possible legitimate doubts of some readers, the presented results are no mere curiosity insofar as the time-honoured Karamata theory is so spread in applied mathematics that it would be unfair to discard the endeavor of constructing a Karamata higher-order theory paralleling the already-constructed one with its present and future applications. In fact, it will be shown that nothing is lost in using either Karmata’s or Hardy’s approaches, so justifying the “only one way” mentioned in the abstract. Of course, this conclusion refers to the chosen procedure in constructing the higher-order theory repeatedly using the standard differential operator d / d x . An approach based on more general operators of type, say,

with suitable regularity conditions on the strictly-positive functions p i would in principle be possible as far as the basic results are concerned, but, as the author’s first impression, it might cause tremendous trouble in developing the “algebra” of the involved functions, e.g., the properties of composite or inverse functions. But this is another story.

This paper concludes the cycle of articles devoted to the general theory of higher-order types of asymptotic variation.

EssentialNotations:

ℝ : = real line; ℝ ¯ ≡ extended real line : = ℝ ∪ { ± ∞ } ; f ∈ A C 0 ( I ) ≡ A C ( I ) ⇔ f is absolutely continuous on each compact subinterval of the interval I ; f ∈ A C k ( I ) ⇔ f ( k ) ∈ A C ( I ) ;increasing ≡ nondecreasing; decreasing ≡ nonincreasing;In the sequel, all the asymptotic relations between strictly positive functions are meant as x → + ∞ :

f oscillatory as x → + ∞ means that f vanishes on each interval [ T , + ∞ ) ;A property holds “ultimately” if it holds true on some interval [ T , + ∞ ) .

This section contains a quick summary of the various concepts of regular and rapid variation as the independent variable tends to + ∞ disregarding a thorough discussion on the historical approaches to the more general concepts of types of asymptotic variation. Each of the involved functions is defined on some interval [ T , + ∞ ) .

So far, we pointed out five different notions of regular variation at + ∞ and are now going to make explicit their mutual relationships. For convenience in the present context, we use notations for the various classes not common in the current literature.

Each of the involved functions is assumed defined on some interval [ T ,+ ∞ ) and strictly positive thereon.

Hardy ’ s classes. ℋ α ( + ∞ ) is the family of functions f ∈ A C [ T , + ∞ ) and such that

Karamata ’ s classes. K α ( + ∞ ) is the family of functions f measurable on [ T , + ∞ ) and such that

Bourbaki-Dieudonn é ’ s classes. ℬ α ( + ∞ ) is the family of functions f such that

or, equivalently, the estimates in (2.13) and wherein no regularity property of f is required a priori.

Zygmund ’ s classes. Z α ( + ∞ ) is the family of functions f measurable on [ T , + ∞ ) and such that

Theorem3.1(Inclusion relations between these classes). The following facts obtain.

(I)

and K α ( + ∞ ) happens to be the algebraic closure of ℋ α ( + ∞ ) with respect tothe equivalence relationof “same growth-order at + ∞ ” defined in (1.3).

(II) If f ∈ A C [ T , + ∞ ) and if f ' is monotonic,then

hence,functions in the set { K α ( + ∞ ) \ ℋ α ( + ∞ ) } ∩ A C [ T , + ∞ ) must have an(ultimately)non-monotonic derivative.

Proof. (I). First, the asymptotic relation in (2.5), valid for the functions in K α ( + ∞ ) , as remarked at the end of Proposition 2.3, shows that K α ( + ∞ ) ⊂ ℬ α ( + ∞ ) .

Second, the strict inclusion is not due to the simple fact that the functions in ℬ α ( + ∞ ) have no a-priori regularity restrictions, unlike the measurability of those in K α ( + ∞ ) , but is a substantial property as stated by the strict inclusion in the second line in (3.5) and proved, e.g., by the function

which is in ℬ α ( + ∞ ) for any ϕ but is in K α ( + ∞ ) only in the very special case that “ 2 + sin ϕ ( λ x ) ~ 1 ,   x → + ∞ , ∀   λ > 0 ”, “ ϕ ( x ) = constant = ( 3 π / 2 ) + 2 k π ”.

The second inclusion in the first line in (3.5) is obvious, whereas the third inclusion, as well as the last claim in part (I), is proved by comparing the representations in (2.4) and (2.18). To visualize this strict inclusion, the reader may check this fact in the following case:

wherein the quantity x f ′ ( x ) / f ( x ) = α + ( x cos x + 1 ) / ( log x + sin x ) has no limit in ℝ ¯ as x → + ∞ . The last equality in (3.5), ℋ α ( + ∞ ) = Z α ( + ∞ ) is in Proposition 2.4.

(II). The exact meaning of the monotonicity assumption, explained in ([3], §4), is that f is either concave or convex on [ T , + ∞ ) so that both one-sided derivatives exist as finite numbers on [ T , + ∞ ) and are monotonic thereon. A proof of the equivalence in (3.6) is in ([3], Th. 4.1).

By Theorem 3.1, we have three non-equivalent notions of regular variation at + ∞ , with a fixed index α ∈ ℝ , highlighted by the three classes ℋ α ( + ∞ ) , K α ( + ∞ ) , ℬ α ( + ∞ ) . In principle, regular variation of higher order is defined by imposing the corresponding condition on some of the derivatives, while suppressing the conventional condition of positivity. The essential features are that the involved functions maintain a constant nonzero sign and exhibit absolute continuity of the appropriate order. For the label concerning the number of the order, we choose the order of the highest involved derivative. The case n = 2 suffices to point out the phenomenon.

Definition4.1Let α be a real number.If in the definitions of the three mentioned classes,each element is assumed to preserve a constant nonzero sign for all x large enough,then we have the three enlarged classes of:

noticing that in the first of these classes the restriction of“nonzero sign”refers to the given function f and not to f ′ as in the next concepts.

Imposing conditions on one further derivative,we define the following classes:

(I)

(II)

Warning : f ′ ( λ x ) stands for f ′ ( y ) | y = λ x and not d d x f ( λ x ) !

(III)

To go on it is necessary to know a link between the numbers α , β : a relationship well-known for the class { ℋ α ( + ∞ )   of   order   2 } .

Theorem4.1If α ≠ 0 then β = α − 1 in each of the three classes described in Definition4.1-(I)-(II)-(III),whereas possible exceptions may occur in the case α = 0 .More precisely,for the three mentioned classes,the following inferences obtain:

Proof. (I) For the class { ℋ α ( + ∞ )   of   order   2 } , the result is known: [1], Prop. 2.6.

(II) For the class { K α ( + ∞ )   of   order   1 } , if α ≠ 0 , then the inequalities in (2.6), granted by Proposition 2.3, together with the constant sign of f ( x ) imply that “ lim x → + ∞ f ( x ) = either 0 or ± ∞ ” so that the pertinent L’Hospital rule may be applied in evaluating the limit:

and the same argument is obviously valid in the case α = 0 under the explicit additional restrictions on f ( + ∞ ) . For the last claim in (4.4), assume, if possible, “ α = 0 , β > − 1 ” which implies “ x | f ′ ( x ) | ∈ K β + 1 ( + ∞ ) ”, whence, in sequence:

But we have just proved that the last property, for α = 0 , implies β = − 1 : a contradiction.

(III) For the class, { ℬ α ( + ∞ )   of   order   1 } the assumptions imply:

and suppose, if possible, that “ 1 + β − α > 0 ”. This would imply lim x → + ∞ ( x | f ′ ( x ) | f ( x ) ) = + ∞ , while an application of the pertinent L’Hospital rule would yield a contradictory result:

Hence, 1 + β − α ≤ 0 . □

Counterexamples for the cases α = 0 . Consider the three functions, reported from [1], formula (2.107):

The following circumstances occur as x → + ∞ :

Next results in the paper—Theorem 4.2 and §5—will show that practically equivalent higher-order theories of asymptotic variation can be built using either Karamata’s or Hardy’s approaches, whereas applications of the classes { ℬ α ( + ∞ )   of   order   n ≥ 1 } would offer less precise asymptotic results.

Theorem4.2 (Karamata’s and Hardy’s higher-order theories).

(I) The following inclusions obtain for each fixed α ≠ 0 :

(II) Let α = 0 andfsatisfy the conditions in(4.2)with some number β ≤ − 1 .If β < − 1 then f ∈ { ℋ γ ( + ∞ )   of   order   1 } where γ equals either ( β + 1 ) or zero.If β = − 1 then f ∈ { ℋ 0 ( + ∞ )   of   order   1 } .

(III) Hence,each function in { K α ( + ∞ )   of   order   1 } , α ∈ ℝ ,belongs to a class { ℋ γ ( + ∞ )   of   order   1 } with a suitable value of γ ,and this fact implies that nothing is lost in using higher-order classes of regular variation built accordingto eitherKaramata’s or Hardy’s approaches.

The proof depends on the following

Lemma4.3 (Regular variation of antiderivatives). Let g ∈ K γ ( + ∞ ) .

(I)

(II)

Proof of Lemma 4.3. (I) This is a classical result by Karamata ([7], pp. 26-27).

(II). These claims are cited by Seneta ([8], Exercises 2.1-2.3, pp. 86-87), but the proof of the inference in (4.15) goes back to a paper by Parameswaran ([9], Lemma 1, pp. 220-221). For the reader’s convenience, we report the proof of (4.15) and its adaptation to (4.16) because Theorem 4.2 is the main result in this paper. It is much more convenient to use different notations putting g ( x ) = L ( x ) x − 1 , with L (strictly positive) slowly varying, and estimating the pertinent antiderivative of g . If k is an integer ≥ 2 , then for (4.15) we have

where the uniformity of the lim x → + ∞ L ( λ x ) / L ( x ) = 1 implies that inside the integral we have the estimate “ L ( t ) / L ( x ) > 1 / 2 ” choosing x large enough, say x ≥ T k , so that

and the arbitrariness of k implies that lim x → + ∞ M 1 ( x ) / L ( x ) = + ∞ which, due to the strict positivity of the involved functions, is equivalent to our assertion. To prove (4.16), we have analogously:

whence lim x → + ∞ M 2 ( x ) / L ( x ) = + ∞ . □

Proof of Theorem 4.2 (I) By Theorem 3.1, the only inclusion to be proved is

which may be proved using the basic result in Lemma 4.3. Let f satisfy the conditions in (4.2) with β = α − 1 . For α > 0 , β > − 1 , we apply (4.13) to f ′ so getting “ ∫ + ∞ | f ′ | = + ∞ ” whence “ f ( + ∞ ) = ± ∞ ” and

For α < 0 , β < − 1 , we have

(II) If α = 0 and β < − 1 we have by (4.14):

and must separate the two circumstances that the number f ( + ∞ ) be zero or not. In the case f ( + ∞ ) = 0 , we are in the situation of (4.21) wherein the second relation in the second line is granted by (4.16) with α replaced by β + 1 , hence f ∈ { ℋ β + 1 ( + ∞ )   of   order   1 } . In the case “ f ( + ∞ ) = ℓ ≠ 0 ”, we have

hence, f ∈ { ℋ 0 ( + ∞ )   of   order   1 } . It remains the case “ α = 0 , β = − 1 ” wherein Lemma 4.3-(II), applied to the derivative, will be used once again.

First circumstance: ∫ + ∞ | f ′ | = + ∞ . This implies:

whence x f ′ ( x ) = o ( f ( x ) ) , i.e., f ∈ ℋ 0 ( + ∞ ) .

Second circumstance: ∫ + ∞ | f ′ | < + ∞ :