The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions . Part I : Higher-Order Regular , Smooth and Rapid Variation

Motivated by a general theory of finite asymptotic expansions in the real domain for functions f of one real variable, a theory developed in a previous series of papers, we present a detailed survey on the classes of higher-order asymptotically-varying functions where “asymptotically” stands for one of the adverbs “regularly, smoothly, rapidly, exponentially”. For order 1 the theory of regularly-varying functions (with a minimum of regularity such as measurability) is well established and well developed whereas for higher orders involving differentiable functions we encounter different approaches in the literature not linked together, and the cases of rapid or exponential variation, even of order 1, are not systrematically treated. In this semi-expository paper we systematize much scattered matter concerning the pertinent theory of such classes of functions hopefully being of help to those who need these results for various applications. The present Part I contains the higher-order theory for regular, smooth and rapid variation.


Introduction
In a previously-published series of papers ( [1] and for practical applications it is quite useful to have some information on the asymptotic behavior of the ratio of Wronskians.For 0 x = +∞ a possible step consists in writing and then recalling that in various but related contexts the following remarkable asymptotic relations linking the ratios ( ) ( ) ( ) are found: for different classes of functions.Bourbaki ([6]; Chap.V, appendix, pp.V.36-V.40), in the context of Hardy fields, shows their validity for each k ∈  (some exceptional cases apart) for the classes of functions therein called "of finite order α [respectively, of infinite order] with respect to the function ( ) : g x x = , as x → +∞ " and defined by the property that ( ) lim log log exists in .
All these approaches for infinitely-differentiable functions have in common the existence of the following limit 0 slow variation at , lim \ 0 regular variation at , rapid variation at , wherein the three contingencies are special cases of the more general classes of functions traditionally labelled as "slowly, regularly or rapidly varying at +∞ ".Motivated by the fact that the ( )   k i φ 's of most asymptotic scales found in applications belong to one of these classes and that the before-mentioned relations for the derivatives have important applications in several fields, we deemed it convenient to systematize the theory of higher-order "types of asymptotic variation" showing the equivalence of various approaches, putting together a large amount of basic properties and highlighting the parallel theory of rapid or exponential variation always cursorily treated.Many proofs have an elementary character left apart: the equivalence of the various approaches based on a remarkable device by Balkema, Geluk and de Haan, and the operations on higher-order varying functions which requires a certain amount of patience.Much time has been spent in giving an abundance of counterexamples to show the necessity of possible restrictive assumptions.A special attention has been paid to listing a variety of asymptotic functional equations satisfied by the functions in the studied classes.Only the general theory has been treated in this semi-expository paper and the applications are restricted to some asymptotic properties of antiderivatives and sums, and to asymptotic expansions of an expression of type ( ) ( ) Applications to general asymptotic expansions and differential-functional equations would require a separate treatment.The exposition is on a plain level and an effort has been made to look for the simplest proofs.
- §2 contains a detailed and integrated exposition of basic properties (algebraic, differential and asymptotic) concerning regular and rapid variation in the strong sense.
Much, but not all, the material concerning regular variation is standard and the most elementary proofs have been reported.Some facts concerning the index of variation of the first derivative in §2.3 are essential both to give a correct definition of higher-order regular variation and to understand possible restrictions on the indexes.

A. Granata
-In §3 we give an integrated exposition of higher-order regular variation (a concept indirectly encountered in the context, e.g., of Hardy fields) and smooth variation (a concept explicitly present in the literature concerning some applications of regular variation), both traditionally (but not in our approach) referred to C ∞ -functions.We show the equivalence of different approaches found in the literature reporting a clarified version of a non-trivial characterization by Balkema, Geluk and de Haan trying to highlight the computational ideas in the ingenious proof, somehow hidden in the original exposition.
-In §4 an analogous exposition for higher-order rapid variation is given with several characterizations.To be useful for applications a restriction must be added to the "spontaneous" concept of higher order for this class of functions.
-In §5 there is a discussion about various useful asymptotic functional equations satisfied by the functions in the previously-studied classes.
In part II we exhaustively describe results about algebraic operations on higher-order asymptotically-varying functions and treat concepts related to exponential variation and some of their basic applications.-The logarithmic derivative ( ) -Hardy's notations: " which we label as "asymptotic similarity", means that -The relation of asymptotic equivalence: -When describing properties related to exponential variation it is convenient to use the following nonstandard notation: and a similar definition for notation We shall formally use these notations like the familiar " ( ) ; if , 0.
where k α is termed the "k-th falling (≡decreasing) factorial power of α ".Notice that we have defined 0 0 : 1 = , hence a linear combination such as ( ) whatever the i α 's.Propositions are numbered consecutively in each section irrespective of their labelling as lemma, theorem and so on.
Notations for iterated natural logarithms and exponentials , defined for large enough ; : ; The special definitions for 0  and 0 exp are agreements.Their derivatives are:

The Elementary Concept of "Index of Variation" and Properties of Related Functions
The general theory of finite asymptotic expansions we constructed in the cited papers essentially deals with functions of the regularity class In the modern well-developed-and-organized theory of regular or rapid variation, with its many applications to probability and statistics, the approach via (1.10) is of secondary importance but for higher-order variation the "natural" approach is that of introducing n C -functions whose all derivatives have an index of variation just in the sense of (1.10); and it will be seen that an additional condition is required for rapid variation.Both for applications and for further theoretical results we need many of the standard properties of regularly-or rapidly-varying functions and so we cannot help giving an almost complete list of them though their proofs are usually elementary even not always obvious; not all of those in-A.Granata volving rapid variation are to be found in texts on the subject.A special attention is given to linear combinations of asymptotic scales.As concerns higher-order variation the essential fact that the classes of higher-order regularly-or rapidly-varying functions are closed with respect to the operations of product, composition and inversion requires nontrivial proofs reported in Part II.The results will make the reader feel quite at ease with the many examples scattered in our work.The asymptotic relations for the ratios ( ) ( ) ( ) φ φ obviously are those familiar in the context of Hardy fields but our context is more general and some useful points about the indexes are highlighted in certain exceptional cases.
Unlike the traditional concept of "order of growth" which involves one specified comparison function we use the generic locution of "type of growth", or better "type of asymptotic variation", to denote one of the classes of functions which are either regu- larly or smoothly or rapidly or exponentially varying; and these are classes which in our exposition are defined via "asymptotic differential equations" whereas for order 1 they may be included in larger classes defined through "asymptotic functional equations".

The Elementary Concept of "Index of Variation"
for some constant α ∈  which is called the index of regular variation of f at +∞ .We denote the family of all such functions for a fixed α by ( ) α +∞  .In the case 0 α = the function f is also termed "slowly varying at +∞ (in the strong sense)". (II) f is termed "rapidly varying at +∞ (in the strong sense)" if Accordingly, the index of rapid variation at +∞ is defined to be either +∞ or −∞ and the corresponding families of functions are denoted by (III) f is said to have an "index of variation at +∞ in the strong sense" if the following limit exists in the extended real line: with the tacit agreement that the limit is taken for x such that ( ) exists as a finite number.Whenever there is no need to specify the index of variation we denote the class of all such functions by the symbol We sometimes omit the specification "in strong sense" as this is the only meaning we are using for this concept.
Remarks. 1. Condition "f ultimately of one strict sign" is essential both in the general and in our restricted definition.The choice +∞ for some T and ( ) 0 f x ≠ for x large enough".However in some cases the positivity of f may be essential for a correct result as when investigating the possible variation-properties of a linear combination.
2. The locution "in strong sense" is a reminder of the fact that our class of functions is a proper subset of the class of regularly-or rapidly-varying functions in more general senses.The first larger class is that of those real-valued functions f defined on a neighborhood of +∞ and admitting of an "order α", with respect to the comparison function ( ) : .
In the monograph [8] our restricted class for α ∈  is called of the "normalized regularly-varying functions" and shown to coincide with the "Zygmund class" ( [8]; pp.15, 24) of those positive, measurable functions f such that, for every is ultimately increasing" and " is ultimately decreasing".The Karamata class of regularly-varying functions at +∞ coincides with the larger class of functions , each defined on some neighborhood of +∞ and asymptotically equivalent to a function regularly varying in the strong sense and we mention in passing that a convex f satisfying (2.6)The specification "at +∞ " is not superfluous; the change of variable x → .The restricted classes we have just defined are appropriate to define the concept of "variation of higher order" and suffice for many applications in the field of ordinary differential equations and asymptotic expansions.To visualize, notice that all infinitely-differentiable functions which can be represented as linear combinations, products, ratios and compositions of a finite number of powers, exponentials and logarithms as well as their derivatives of any order have principal parts at +∞ which, as a rule, can be expressed by products of similar functions, hence such functions are strictly one-signed, strictly monotonic and strictly concave (or convex) on a neighborhood of +∞ so that the quantity ( ) ( ) is ultimately monotonic and the limit in (2.3) is granted.
3. Typical (indeed the most usual and useful) functions in , , , , , whose index of variation is: " ( ) γ may be any number >0.Also notice that " ( ) -Typical monotonic functions in ( ) ) and their products ) Separating the cases 0 α ≠ and 0 α = we may rewrite (2.1) in the form ( ) each of these may be viewed as an "asymptotic (ordinary) differential equation of first order" and it is easily shown (Proposition 2.1 below) that the solutions of the first one of them share the asymptotic properties of the solutions of the ordinary differential equation ( ) ( ) we get the characterization: An absolutely continuous function f belongs to the class , iff there exist two numbers such that f admits of the following representation: And an analogous statement holds true for a rapidly-varying function with ( ) As a first rough asymptotic information: Notice that either representation " ( ) ( ) The general classes of regularly-or rapidly-varying functions enjoy many useful algebraic and analytic properties but it is not self-evident that the same is true for our restricted classes, in particular that they are closed with respect to various operations.In the next subsection we give a list of the main properties omitting those proofs which are quite elementary based on the property of the logarithmic derivative: . (2.17)

Basic Properties of Regularly-or Rapidly-Varying Functions
Proposition 2.1.(Algebraic and asymptotic properties of regularly-varying functions).
The following properties hold true: (i) Factorization: (ii) Growth-order estimates: But for 0 α = all the possible contingencies may occur for this limit as shown by the following functions of class ] ) exp log sin log , 0 1 2.
The third and the fourth of these functions are not ultimately monotonic: The third with bounded oscillations and the fourth, call it f, with unbounded oscillations: ( ) then the following functions are regularly varying as well with the specified index of variation δ : ( ) ( ) with index max , constant 0 .
For α β ≠ no restriction on the signs of the i c 's is necessary in (2.27), obviously not both zero.
(iv) Composition.If , f g are as in (iii) and if ( ) above list we may add the composition In particular, if ( ), In the case 0 α = it may happen that log f has no index of vari- ation as shown by the third function in (2.22) for some values of the constant c: In fact as x → +∞ the function changes sign infinitely often for 1 2 c < < , has infinitely many zeros for and it is slowly varying for The particular case in (iii) and (iv) states that if , f g are slowly varying then so are the functions , ; ; ; , constant 0 ; , constant 0 ; , provided that .
The examples in (2.22) show that a pair ( ) may not be comparable at +∞ meaning that one or both of the limits " lim , lim f g g f , as , x → +∞ " may fail to exist in  .By the factorization in (2.18) the same applies to a pair ( ) , then f ′ has ultimately one strict sign hence the restriction of f to a suitable neighborhood of +∞ has an inverse function defined on a neighborhood of +∞ as well and we have that ( ) ( ) whereas no inference can be drawn if α β = as shown by the functions in (2.22).Proof.(i) is trivial and (ii) follows from (2.13) as ( ) For the linear combination in (2.27) in the case α β = we have: by the positivity of , .
To prove (vi) evaluate the following limit by the change of variable ( ) 2. In (2.27) it is essential that all the involved quantities (functions and constants) have one and the same sign for α β = otherwise possible cancellation of terms may yield any growth-order as shown, e.g., by ( ) ( ) where φ is any of the three functions , 0;e ; sin , 0 3. The "Zygmund property" cited after (2.6) and concerning the ultimate strict monotonicity of is trivially checked for regular variation in the present strong sense by directly evaluating the derivative of this product and using (2.11).

4.
A less direct proof of (2.27) uses the decomposition: Examples.1. Referring to the third function in (2.22) we mention that it can be proved that the function " even in the general weak sense and the same is true for " ( ) " is regularly varying (of index α ) in the general weak sense for each 0 α > for the simple reason that " sin ~, x x x x α α + → +∞ "; but it is regularly varying in the strong sense only for , is slowly varying and tends to zero, as , x → +∞ faster than any negative power of log x ; and the function The following properties hold true: (i) Growth-order estimates: , , > 0, hence 0. .
with no inference about the quotient f g in .Together with the result in (2.25) we may assert that: For any two functions " with no inference about f g e log ; log exp ; log e .exp In particular Roughly speaking the inverse of a rapidly-varying function is slowly varying and viceversa.To be precise, if ( ) Proof.For the first three groups of relations we write down the proof only for ) In the next proposition we collect various results about linear combinations, results particularly useful in asymptotic contexts.( ) , , ; without any further restriction on , f g .On the contrary, we can prove the following two inferences only under one of the two specified restrictions: , ; provided that: These inferences, together with (2.27), are summarized in whatever the positive constants 1 2 , c c and the extended real numbers , α β ∈  ex- cept for the two cases in (2.53) wherein a restriction is added: See a discussion after the proof.
(II) (Arbitrary linear combinations of asymptotic scales).Let the functions , , and let one of the following conditions be satisfied, either ( ) , , 2; in particular , ; or ( ) , , 2; in particular , ; with ( ) ( ) so that: if 1 φ has an index of variation at +∞ in the strong sense also the function ( ) has the same index of variation.Without both additional conditions (2.57)-(2.58)there is no general claim about the type of growth of i i i c φ ∑ as shown by simple counterexamples reported after the proof.
, , 0; , , , , Proof.We may include the constants i c into the functions , f g .For the first two inferences in (2.52) the assumptions imply that respectively for the first and the second inference and the claims follow.For the third inference in (2.52) we have " ( ) A different elementary proof is achieved writing: ; and this is a special case of the result in part (II).The two inferences in (2.53) follow from part (II) under condition ( ) . Under condition " f ′ ultimately monotonic" an argument runs as follows: the assumptions in both cases imply " ( ) ( ) . Moreover it will be proved in Proposition 2.5 that the monotonicity of f ′ implies its satisfying the same asymptotic estimates as f in (2.41) i.e. " ( ) ( ) ∀ > ".Now we have: because we shall prove in a moment that " ( ) ( ) ( ) In part (II) the result involving (2.57) trivially follows from factoring out 1 1 c φ and c φ′ in the left-hand side in (2.59), whereas to prove (2.59) under (2.58) we have first to notice that ( ) ( ) .
Claims in parts (III), (IV) are corollaries of the result in part (II) involving (2.58). Remarks. 1.Using the decomposition in (2.39) the first two inferences in (2.52) may be proved with the restriction " ( ) The trivial device in (2.66) would work well also under the assumptions: (which grants 0 g′ ≠ ).This last restriction is overcome in the different proof provided for (2.27) as well as in an alternative more elaborated proof involving decomposition (2.39).
2. Conditions in (2.57) and (2.58) are independent.Any pair f, g where f is any function of type in (2.7) and g is any function of type in (2.8) with 2 0 c < , hence ( ) : and this shows that lim inf lim inf 0; lim sup lim sup ; ~~1 hence , .
x x Counterexamples concerning suppression of conditions (2.57)-(2.58).In the following we use three pairs of functions in ( [4]; (9.12), (9.13), (9.14); p. 487): In the last example the function For integrals of functions in our classes we report the classical results (with elementary proofs) to highlight a difference between the two cases.x f f The inferences in (2.74)-(2.75)are respectively equivalent to the following asymptotic relations expressing the behavior of the integral In the two cases . ., ; . ., .
But under the stronger assumption we have the exact principal parts: In the second and fourth cases one applies L'Hospital's rule to ( ) ∫ , preliminarly noticing that ( ) ( ) with a suitable constant c.Using the third condition in (2.83) we get Remarks.1.There is a difference between the two cases: though the character of regular or rapid variation of an antiderivative is elementarily checked, a useful result about the asymptotic behaviors in the rapid-variation case requires a restrictive assumption.
The following counterexample shows that it is not easy to get rid of a condition like this even if the second condition in (2.83) is replaced by the stronger one

89)
Here: ( ) ( ) 2. The proof based on the device in (2.87) could be adapted to the case of a regularly-varying function observing that, for 0 α ≠ , (2.1) is equivalently expressed as and imposing the extra-assumption of formal differentiation of this last relation, i.e.
( ) ( ) ( ) One would re obtain the inferences in (2.74)-(2.75)but under the unnecessary restrictions: (2.91) and 0 α ≠ .Hence the device in (2.87) is unnatural for a regular- ly-varying f whereas it is appropriate to the rapidly-varying case.

Properties of the First Derivative
The following properties of the first derivative are essential to develop the theory of higher-order variation.
Proposition 2.5.(Elementary asymptotic properties of the first derivative).The following hold true with all asymptotic properties referring to x → +∞ .
(I) (Regular variation).The estimates in (2.19) imply that: : 2 sin log , 0 1 , 2 sin log 1 , bounded but nonconvergent; Moreover, for each 0 α ≠ we can exhibit an " ( ) " such that f ′ is not ultimately monotonic.In both the following examples the reader will check the asymptotic formulas for f ′′ showing that f ′′ changes sign infinitely often and that f ′ has no index of variation: and f ′ may be ultimately monotonic, as for the functions in (2.9) or it may even change sign infinitely often as for the functions " ( ) (III) (Rapid variation).
( ) ( ) ( ) ( ) the additional condition " f ′ monotonic" grants that f ′ satisfies the same asymptotic estimates as f : ( ) We do not know if relations in (2.102), or even the simple relation Proof of part (III).The estimates for f ′ in (2.100) follow from where both factors tend to +∞ by (2.41).This simple argument does not work for ( ) A. Granata but in this case the limit " ( ) ( ) " does not exist as well.
As far as the possible index of variation of the first derivative is concerned notice that if ( ) f ∈ +∞  then f ′ may have no index of variation at +∞ as shown by the following counterexamples where the term "oscillatory" means that the pertinent function changes sign infinitely often as x → +∞ : 2e sin e , oscillatory and unbounded ; e e sin e , oscillatory and unbounded .
But if f ′ has an index of variation then there are precise important links between the two indexes.The results in the next proposition are essential in the higher-order theory and to understand why restrictions on the indexes are sometimes required.
Proposition 2.6.(Index of variation of the first derivative).(I) If [ ) and if both f and f ′ have indexes of variation at +∞ , respectively α and α ′ , then: { } ( ) In the case 0 α = and without the stated additional condition on ( ) but it cannot be 1 α ′ > − .Hence for α ∈  it always is: for some γ ∈  and in such a case it is necessarily ( ) The same argument is valid for 0 α = and the stated restriction on ( ) f +∞ .It re- mains the case " α = −∞ " which implies " ( ) 0 f +∞ = " and this condition leads to ex- cluding the following contingencies for the indicated reasons: (i) α ′ = +∞ would imply " ( ) = +∞ ", and by L'Hospital: which is a positive real number; hence " = ", and this would imply the contradiction: (iv) The case 1 α ′ = − must be treated in a different way using the estimates in (2.19) and (2.41).In our present proof we have α = −∞ and and there are two a-priori contingencies concerning the integral f +∞ ′ ∫ . Its divergence would imply ( ) f +∞ = +∞ which cannot be; in the other case we would have ( ) contradicting the first relation in (2.114).Notice that the procedure used to prove this last case works for any α ′ ∈  as well.
The last assertion in the statement, namely "it cannot be For part (II) the assumptions for (2.108) are: whence (2.108) follows.Viceversa assume ( ) as the factors on the right diverge either both to +∞ or both to −∞ .

The Concept of Higher-Order Regular Variation
By the foregoing proposition we can define unambiguously some concepts of "higher-order asymptotic variation" separating the cases of regular variation (in this section) and rapid variation (in the next section).
Definition 3.1.(Regular variation of higher order).A function is termed "regularly varying at +∞ (in the strong sense) of order n" if each of the functions never vanishes on a neighborhood of +∞ and is regularly varying at +∞ with its own index of variation according to Proposition 2.6.If this is the case we use notation Whenever needed we denote the indexes of the derivatives as follows: .
Remarks. 1.It is essential to consider the absolute values in order to not impose a-priori restrictions on the signs of the derivatives.Saying that " f is regularly varying of order 1" means that " f is regularly varying in the sense of Definition 2.1".The functions in (2.7) are regularly-varying at +∞ (in the strong sense) of any order n.
The index 1 α was denoted by α ′ in Proposition 2.6.
By (2.106) the inference in (3.3) may well hold true without the stated restriction whenever " 0 p α = " for some p and " ( ) ( ) p f +∞ = either 0 or +∞ ".In any case, though not all the indexex k α may be uniquely determined a priori, there are precise and fundamental asymptotic relations linking each ( hold true whichever α ∈  may be.For 0 α ≠ they may be written as if and only if the following relations hold true with suitable constants k γ such that , , 0; no restriction on .
n n If this is the case then: , 1 ( ) Replacing the relation for ( ) ( ) and iterating the procedure yields which by (2.106) coincides with (3.5) under the assumptions in (3.3).Under the assumptions in (3.4) we get relations in (3.5) for In any case (3.5) hold true for 1 k n ≤ ≤ .Relations in (3.6) simply follow from the inference For part (II) we must prove that relations (3.7)-(3.8)imply ( ) ( ) ( ) ( ) ( ) The claim for 2 n = is contained in Proposition 2.6-(II) and we proceed by induction assuming the claim true for a certain with satisfying 3.8 , the inductive hypothesis implies 0 and ~1 for 1 ; and we must prove the relation in (3.7) with k replaced by and replace this expression into the relation in (3.14) for = − this is the thesis without any restriction on n γ ; for . Hence in (3.17) it must be : 2 sin log , , ; and f ′ cannot be regularly varying as it has al- ternating signs.
Second counterexample: ) : e of order 1 and no more , though the relations in (3.7) hold true for each Third example:

The Concept of Higher-Order Smooth Variation
The second counterexample above shows that the set of relations in (3.5) in themselves do not grant that all the involved derivatives be regularly varying: it may well occur an abrupt transition from regular variation to rapid variation at a certain order of derivation.This is the main motivation for our Definition 3.1.But the asymptotic relations for ( ) k f f are most important in applications and in this subsection we report three characterizations of these relations encountered in the literature and valid for any α : the first deals with the derivatives of the ratio in (2.3), ( ) ( ) , and is used in the monograph by Lantsman [9]; the second is a slight variant dealing with the derivatives of the logarithmic derivative ( ) ( ) ; the third highlights the behavior at +∞ of the derivatives of the associated function ( ) ( ) . This last characterization is a nontrivial and useful result proved by Balkema, Geluk and de Haan ( [7]; Lemma 9, p. 410) using an ingenious device.
Proposition 3.2.(Several characterizations).For an large enough, the following four sets of asymptotic relations, for a fixed α ∈  , are equivalent to each other: The reader will notice in the proof that the differential expressions " stem out from successive differentiations of ( ) Proof.We use notation . To prove "(3.22) ⇔ (3.23)" we use the identity from which, putting 0 k = , the equivalence easily follows for 1 n = .By induction suppose the equivalence true for a certain the sum in square brackets being 0 ≡ .Let us now consider the obvious relations concerning the function φ defined in (3.24) and valid for all The equivalence (3.22) ⇔ (3.24) is contained in the following: First, it is elementary to prove by induction the formula wherein we have used the expression of f ′ to get the final expression of f ′′ and the expression of f ′′ to get the final expression of f ′′′ .It is clear that further differentia- tions yield expressions for the operators ( ) ( ) ( ) ( ) and we shall prove the following representation: If now (3.34) is assumed true for a certain k then, differentiating both sides and using (3.35) in the left-hand side, we get where we have put we may write by (3.21): To prove the first inference in (3.45) put ( ) ( ) . Recalling that f has ultimately one strict sign and that " ( ) and by (3.21) we have for each A similar proof in case of divergence.Notice that, with φ defined in (3.24), f is recovered by the formula ( ) ( )  and that regular variation of φ may have ambiguous effects on f .The reader may check that: " and the estimates in (2.19) referred to φ .The case 1 α = remains undecided as shown by: : log , ; ~; hence : , ; for some unspecified constants k γ .Then " and, moreover, for 2 n = there is nothing further to be proved.On the contrary if for 3 n ≥ some coefficient k γ is zero then our claim will be proved once we show that all the successive coefficients are zero as well.In fact if " 2 , , 0 1, ; and the proof is over.Note in passing that the second circumstance implies " ( ) ( ) " as the two relations in (3.53) imply:

The Theory of Higher-Order Rapid Variation
Before giving the proper definition of higher-order rapid variation it is good to add some remarks about the additional condition

F x f
= ∫ , with the proper choice of 0 T ≤ +∞ , then the asymptotic behavior of F can be reread as and, changing again notation, we have one of the following two equivalent relations: ~; ~, i.e. ~, .
For some applications conditions like those in (4. If f is rapidly varying at +∞ of order In most cases we shall not be interested in functions satisfying (4.5)- (4.6) but not (4.7), and so we use no additional notation to highlight the "strong restricted sense".
Remarks. 1.According to our definitions when we speak of a function f rapidly varying (without specifying the order) we are using Definition 2.1 meaning that: > for x large enough and (2.2) holds true.But when we speak of a function f rapidly varying of order 1 (usually omitting the additional locution "in the strong restricted sense") we are using the stronger Definition 4.1. .
It follows that even ( ) ( ) for almost all x large enough.Relations in (4.10) are formally obtained from those in (3.6) as the index α tends to +∞ .Relations in (4.10) imply the following asymptotic scale: whereas a different way of writing relations in (4.6) would give the weaker scale: , .A remark about the ratios ( ) ( ) h k f f .In Definition 4.1-(II) relations in (4.6) imply the following chain: .
A remark about ( ) holds true for some real number a and if it is known that ( ) , then it must be 0 a = .Analogously, using the identity in (4.15) and under the regularity assumptions in Proposition 4.1, we prove that if

Asymptotic Functional Equations for Regular
or Rapid Variation [2] [3] [4][5]) we established a general analytic theory of finite asymptotic expansions in the real domain for functions f of How to cite this paper: Granata, A. (2016) The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions.Part I: Higher-Order Regular, Smooth A. Granata one real variable sufficiently regular on a deleted neighborhood of a point 0 x ∈  .The theory is based on the use of a uniquely-determined linear differential operator instance one such condition is and de Haan, ([7]; Lemma 9, p. 410), have shown the equivalence of (1.4), written in the form

∫
is either convergent or divergent for some T large enough.
right-hand side of (1.3) for functions with such regularity with no additional restrictions either of algebraic character or of C ∞ -regularity such as in the theory of Hardy fields, Bourbaki ([6]; Chap.V, Appendix), or in other expositions, Lantsman ([9]; Chap.5).
of the corresponding classes as 0 x 38) Last: (2.32) follows from (2.25) and (2.21); relations in (2.31) follow from (2.32).Remarks. 1.The properties in (iii), (iv) and (vi) are the same as those valid for the standard powers.The first inference in (2.31) can be interpreted by saying that the class closed under the relation of "asymptotic equivalence" in the specified sense that any regularly-varying function asymptotically equivalent to a function in ( ) α +∞  belongs to the same class; but it is false that "any function in [ ) , AC T +∞ and asymptotically equivalent to a function in ( ) α +∞  is in the same class" for the simple reason that it may not be regularly varying in the strong sense: counterexample of ( ) : This shortcoming is overcome by the Karamata con- cept of regular variation.
39)A.Granata similar to a device which reveals efficient in the case of slow variation in the general weaker sense: see Seneta ([10], p. 19).
is slowly varying and diverges to +∞ faster than any positive power of log x .Proposition 2.2.(Algebraic and asymptotic properties of rapidly-varying functions).
of index 2 in Karamata'e sense.3.As concerns an additional restriction in the two inferences in (2.53) we have already remarked that the assumptions imply " f g g + ", hence if f g + has an index of variation at +∞ this must equal the index of g.We do not know whether inferences in (2.53) can be proved without any restriction or not.A possible counterexample with on the same calculations in (2.67) would be provided by a function f such that:But for the time being we do not know any such function: see Proposition 2.5-(III).

Proposition 2 . 4 .
(I) (Integrals of regularly-varying functions).Let (2.76)-(2.77)we may only assert, generally speaking, that (Integrals of rapidly-varying functions).We have the rough estimates: .85) Remarks.Notice that the formal rule in (2.84)-(2.85)does not coincide with that in (2.78)-(2.79) in accordance with relations in (2.104) below.If an antiderivative of its then the ( ) ( ) lim x F x f x →+∞ may have any value depending on the behavior of ( ) D f x  , and this gives rise to three concepts of "exponential variation" studied in Part II, §8, of this paper.Proof.The convergence or divergence of the integral in case 1 α ≠ trivially follows from the estimates in (2.19).(I) In the first and third cases one applies L'Hospital's rule to x appears on the left in(2.103).The summability of f ′ in (2.101) simply follows from ( ) 0 f +∞ = .The esti- mates in (2.102) follow from those in (2.41) by a classical result which requires either " f everywhere differentiable and f ′ monotonic" or " ": see[12] both for historical references in the introduction and for generalizations.Notice that it is easy to give an example of a function g f such that +∞ does not exist in  with g′ either bounded or not: − as shown by the simple examples: 2.110)    and we do not know whether the partial converse holds true i.e. if both conditions " of part (I) is taken from ([13]; proof of Lemma 2.3, p. 111]).By hypothesis the following two limits exist in  :We now evaluate α by L'Hospital's rule noticing that: " 0 α < ≤ +∞ " implies " ( ) f +∞ = +∞ ", whereas " 0 α −∞ < < " implies " ( ) 0 f +∞ = " and the first limit in (2.111) implies " In both cases the rule may be applied and

1 )
because its derivative has no index of variation; a polynomial of exact algebraic degree 1 n ≥ belongs to the class for some α ∈  then1 n α ≥ − .3.If (3.1) holds true then, by Proposition 2.6-(I): ( )   k fto f for 1 k n ≤ ≤ , and depending only on α .Notice that, by our agreements, a notation like " that the involved derivatives of f are regu- larly varying; for instance it is misleading to write " 1. (Principal parts of higher derivatives in case of regular variation).

Both claims for 2 n 3 n
= are contained in Proposition 2.6-(I), (II): if 0 ≥ we have by assumption the set of relations ( ) is over.As noticed in the proof, (3.5) holds true for any α , hence if the coefficient in the right-hand side vanishes for a certain value of k , say k , it vanishes for all k k ≥ the proof of part (II), cannot be suppressed otherwise any circumstance may occur.First counterexample:

2 n
≥ ; (3.22) true for 1 n + imply the rela- tions in (3.23) true for 1 k n ≤ ≤ whereas (3.25), for k n = , yields implies the relation in (3.23) for 1 k n = + .Viceversa, if (3.23) are true for 1 n + then the relations in (3.23) are true for 1 k n ≤ ≤ whereas, for 1 k n = + , we get from (3.25): explicit expressions of the coefficients i c are not needed except for 1 k c = .If the set of relations on the left in (3.29) holds true then (3.30) at once implies the validity of the set on the right whereas we proceed by induction to prove the converse inference.Let the set of relations on the right in (3.29) holds true with k replaced by 1 k + , then the inductive assumption grants all relations on the left in (3.29).Now we consider (3.30) with k replaced by 1 k + , and solve it with respect to ( ) difficult equivalence is "(3.21)⇔(3.24)" a direct proof of which via Faà di Bruno's formula would involve cumbersome calculations.We report the original proof in a somewhat simplified form explicitly writing the arguments of the involved functions, avoiding the use of a change of variable, and with some additional passages to motivate the technical ideas of the proof.From representation direct routine differentiation and factoring out common factors, we get 3.38)    the right-hand side being a polynomial in and this proves (3.34) for 1 k + .Starting from (3.34) the proof by induction of the equivalence "(3.21) ⇔ (3.24)" is quite trivial.Balkema, Geluk and de Haan ([7]; p. 412) call "smoothly varying of exponent (≡ index) α " a positive C ∞ -function f defined on a neighborhood of +∞ such that the associated function φ satisfies the relations in (3.24) for all k ∈  .In our context we give Definition 3.2.(Smooth variation of higher order).A function enough, is termed "smoothly varying at +∞ of order n and index α " if the four equivalent properties in Proposition 3.2, re- ferred to f , are satisfied.We denote this class by: { ( ) S α +∞  of order n}.Notice that in our definition f is allowed to be either >0 or <0, the essential point being that it ultimately assumes only one strict sign.From Proposition 3.1-(II) and the examples (3.18)-(3.20)we get the following inclusions: .50) To end this section let us ask ourselves what can be said about relations in (3.21) holding true with some unknown coefficients k γ on the right and we give a result- needed in the sequel-concerning the circumstance 1 0enough, satisfy the following asymptotic relations: claim amounts to state that the k γ ' s coincide with the coefficients in(3.21).Now, if " .54) using (3.51) and then the secod relation in(3.53)

2 n
≥ in the previous strong restricted sense then, by (2.106), all the functions f enjoys the properties in (4.5)-(4.6)-(4.7)plus the corresponding value of the limit in (2.2).

2 ."
Conditions in (4.7) obviously imply those in (4.6) whereas, viceversa, complicated calculations in the attempt of proving (4.7) in addition to (4.6) may be usually saved using the classical result (already mentioned in the proof of Proposition 2.5) that: ".For instance (4.5)-(4.7)are trivially satisfied for the functions listed in (2.8) which are the most common functions rapidly varying at +∞ of any order n ∈  in our strong restricted sense.As concerns the analogue of Proposition 3.1 it happens that relations in (3.5) have no analogues for rapidly-varying functions of higher order whereas those in (3.6) have so yielding a useful characterization of this class of functions.Proposition 4.1.(Principal parts of higher derivatives in case of rapid variation).Let in (4.9) and in(4.11)simply are different ways of rewriting relations respectively in (4.8) and in (4.10).Now inspecting (4.7) we have for 1 n = :

7 )+
are equivalent to(4.9).It remains to prove the equivalence between (4.9) and (4.10) for 2 n ≥ and 1 k ≥ .Supposing relations in (4.9) true we start from the asymptotic relation involving( )   in the right-hand side with the analogous relation while leaving unaltered ( ) k f ; so we get

(
reapply the procedure to the last expression in (4.16) so getting gives ( ) ( ) is (4.10) for 2 k = ; and if 3 k ≥ we repeat the procedure and get all relations in (4.10).Viceversa suppose (4.10) true and being bounded.The relations hold true: additional property " ( )f f ′ ′ bounded" is satisfied.For 1 4 c > we also have an example of a function f such that: associated function φ has a second derivative oscillatory and un- bounded.This shows that the properties of the associated function have little meaning, if any, in the context of rapid variation.
, if the asymptotic relation in(4.29) This fact will be needed in the sequel.Corollary 4.2.(Summing up the behaviors of the higher derivatives).Let  .Then, as x → +∞ : is strictly positive, increasing and convex on [ ) 1, +∞ but the thesis in (2.84) does not hold true because the oscillating factors appearing in the expressions of F,F',F" are not comparable between each other though they are very small when compared to the ex- The very same inferences in the case of regular variation, i.e. with  replaced by  , are included in Propositions 2.4-(I), 2.6-(I) and Definition 3.1.
3) are necessary for meaningfulA.Granata general results as, e.g., in determining asymptotic expansions of antiderivatives and in another class of expansions studied in Part II, §11, of this work: this justifies the following restricted concept of rapid variation.
+∞ in the strong restricted sense and this amounts to say that the following conditions hold true as x → +∞ : k assume, by induction, that they imply (4.9), i.e. (4.8), for k ranging in { } An instructive counterexample concerning Definition 4.1 and the associated function.Let us consider the following function