In three previous papers [1] [2] [3] we developed a general theory of higher-order types of asymptotic variation for functions differentiable a certain number of times on some interval [ T , + ∞ ) , spending much effort on results about operations with such classes of functions in [2] and pointing out some elementary applications of the general theory to integrals and sums. Subsequently, in [4] [5], we applied the theory to the difficult problem of evaluating the exact asymptotic behaviors of Wronskians whose entries belong to one or more of the classes of regularly-, smoothly- or rapidly-varying functions. In turn, these results have been applied to the theory of finite asymptotic expansions in the real domain. Going further in this direction, namely trying to evaluate the principal part of certain Hankel determinants, we became aware of some gaps in the long list of results concerning operations with higher-order types of variation: products, linear combinations and inversion. In the present paper, we prove some complementary results that will provide useful in some applications to be developed elsewhere.

For the reader’s convenience we report the general notations and essential facts of the theory, already listed in [ [3]: pp. 435-438] and [ [4]: pp. 6-8].

General notations.

- ℕ : = { 1 , 2 , ⋯ } ; ℤ : = { 0 , ± 1 , ± 2 , ⋯ } ;

- ℝ : = realline ; ℝ ¯ ≡ extendedrealline : = 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 ) ;

- For f ∈ A C k ( I ) we write “ lim x → x 0 f ( k + 1 ) ( x ) ” meaning that x runs through the points wherein f ( k + 1 ) exists as a finite number; f ( + ∞ ) : = lim x → + ∞ f ( x ) .

- The differentiation operators: D f ( x ) : = f ′ ( x ) ; D k f ( x ) : = f ( k ) ( x ) .

- The logarithmic derivative: D l f : = f ′ / f .

- Hardy’s notations:

“ f ( x ) ≪ g ( x ) ,   x → x 0 ” or, equivalently “ g ( x ) ≫ f ( x ) , x → x 0 ” stands for f ( x ) = o ( g ( x ) ) , x → x 0 ;

“ f ( x ) ≼ g ( x ) , x → x 0 ” or, equivalently “ g ( x ) ≽ f ( x ) , x → x 0 ” stands for f ( x ) = O ( g ( x ) ) , x → x 0 .

- The relation of “asymptotic similarity”, “ f ( x ) ≍ g ( x ) ,   x → x 0 ” means that

( c 1 | g ( x ) | ≤ | f ( x ) | ≤ c 2 | g ( x ) |     ∀   x   in   a   deleted   neighborhood   of   x 0 , ( c i = constant > 0 ) . (1.1)

- The relation of “asymptotic equivalence”:

f ( x ) ∼ g ( x ) ,   x → x 0 stands for f ( x ) = g ( x ) [ 1 + o ( 1 ) ] ,   x → x 0 .

- The non-standard notation:

f ( x ) = + ∞ ( g ( x ) ) ,   x → x 0   ( x ∈ I ) , ⇔ def ( f ( x ) = h ( x ) g ( x )     ∀   x   near   x 0 , l i m x → x 0 ,   x ∈ I h ( x ) = + ∞ ; (1.2)

and a similar definition for the notation f ( x ) = − ∞ ( g ( x ) ) ,   x → x 0   ( x ∈ I ) . In particular:

f ( x ) = ± ∞ ( 1 ) ,   x → x 0   ( x ∈ I ) , ⇔ def l i m x → x 0 ,   x ∈ I f ( x ) = ± ∞ . (1.3)

- Factorial powers:

α 0 _ : = 1 ;   α 1 _ : = α ;   α k _ : = α ( α − 1 ) ⋯ ( α − k + 1 ) ;     α ∈ ℂ ,   k ∈ ℕ ; (1.4)

where α k _ is termed the “k-th falling ( ≡ decreasing) factorial power of α ”. Notice that we have defined 0 0 _ : = 1 .

- Everywhere the symbol “ log x ” stands for “ log e ( x ) ”:= “the natural logarithm” of x.

Classes of functions and their main characterizations.

(I) (Index of asymptotic variation). If f ∈ A C [ T , + ∞ ) , f ultimately > 0, its index of asymptotic variation at + ∞ is defined as the value of the following limit (assumed to exist):

l i m x → + ∞ x f ′ ( x ) / f ( x ) = { 0 ( slow variation at + ∞ ) , α ∈ ℝ \ { 0 } ( regular variation at + ∞ ) , ± ∞ ( rapid variation at + ∞ ) . (1.5)

In this case, we use the notation “ f ∈ R α ( + ∞ ) ” with the appropriate value of α ∈ ℝ ¯ .

(II) (Higher-order regular variation). A function f ∈ A C n − 1 [ T , + ∞ ) ,   n ≥ 1 is termed “regularly varying at + ∞ (in the strong sense) of order n” if each of the functions | f | , | f ′ | , ⋯ , | f ( n − 1 ) | never vanishes on a neighborhood of + ∞ and is regularly varying at + ∞ with its own index of variation. If this is the case we use notation

f ∈ { R α ( + ∞ )   of   order   n } ,     α : = “ the   index   of   f ” ∈ ℝ . (1.6)

Notation “ f ∈ { R α ( + ∞ )   of   exact   order   n } ” implies that f is not regularly varying of order ≥ n + 1 .

If f ∈ { R α ( + ∞ )   of   order   n } ,   n ≥ 1 , then relations

( f ( k ) ( x ) / f ( x ) = α ( α − 1 ) ⋯ ( α − k + 1 ) x − k + o ( x − k ) ≡ α k _ x − k + o ( x − k ) ,   x → + ∞ ,   1 ≤ k ≤ n , (1.7)

hold true whichever α ∈ ℝ may be. The indexes of the derivatives are subject to the restrictions specified in [ [1]: Prop. 2.6, p. 796]; in particular:

f ∈ { R 0 ( + ∞ )   of   order   n } ,   n ≥ 2,   ⇒   ( f ′ ( x ) = o ( x − 1 f ( x ) ) , f ″ ( x ) = α 1 x − 1 f ′ ( x ) [ 1 + o ( 1 ) ] with   α 1 ≤ − 1 ; (1.8)

where α 1 is the index of f ′ and the index of f ( k ) is “ α 1 − k + 1 ” for k ≥ 2 .

Notice that the last derivative involved in (1.7), i.e. f ( n ) , may have an arbitrary sign if α n _ = 0 .

The following partial converse of relations in (1.7) holds. If f ∈ A C n − 1 [ T , + ∞ ) ,   n ≥ 2 , then f ∈ { R α ( + ∞ )   of   order   n } for some real number α ∉ { 0,1, ⋯ , n − 2 } iff the following relations hold true:

f ( k ) ( x ) / f ( x ) = γ k x − k + o ( x − k ) ,   x → + ∞ ,   1 ≤ k ≤ n , (1.9)

with suitable constants γ k such that

γ 1 , ⋯ , γ n − 1 ≠ 0 ;     ( no   restriction   on   γ n ) . (1.10)

If this is the case then: γ k = α k _ ,   1 ≤ k ≤ n .

(III) (Smooth variation). The preceding partial converse justifies the following concept.

A function f ∈ A C n − 1 [ T , + ∞ ) ,   n ≥ 1 , f ( x ) ≠ 0 ∀ x large enough, is termed “smoothly varying at + ∞ of order n and index α ” if the relations in (1.7), referred to | f | , are satisfied. We denote this class by: { S R α ( + ∞ ) of order n}. The following inclusions obtain:

( { R α ( + ∞ )   of   order   n } = { S R α ( + ∞ )   of   order   n }                                                                       if   n = 1     or     { n ≥ 2,   α ≠ 0,1, ⋯ , n − 2 } ; { R α ( + ∞ )   of   order   n } ⊊ { S R α ( + ∞ )   of   order   n }     otherwise ; (1.11)

the reason for the last strict inclusion being that some derivatives of a smoothly-varying function may vanish infinitely often or change sign infinitely often. The following sets of asymptotic relations, for a fixed α ∈ ℝ , are equivalent to each other:

x k f ( k ) ( x ) / f ( x ) = α ( α − 1 ) ⋯ ( α − k + 1 ) + o ( 1 ) ,   x → + ∞ ,   1 ≤ k ≤ n ; (1.12)

( x f ′ ( x ) / f ( x ) = α + o ( 1 ) ,   x → + ∞ ; ( x f ′ ( x ) / f ( x ) ) ( k ) = o ( x − k ) ,   x → + ∞ ,   1 ≤ k ≤ n . (1.13)

(IV) (Rapid variation of first order). A function f ∈ A C 1 [ T , + ∞ ) is called “rapidly varying at + ∞ of order 1 (in the strong restricted sense)” if:

( f ( x ) , f ′ ( x ) ≠ 0 ∀   x   large   enough ; f ( x ) / f ′ ( x ) = o ( x ) , x → + ∞ ; ( f ( x ) / f ′ ( x ) ) ′ = o ( 1 ) , x → + ∞ ; (1.14a)

or, equivalently, if:

( f ( x ) , f ′ ( x ) ≠ 0 ∀   x     large   enough ; f ″ ( x ) / f ′ ( x ) ∼ f ′ ( x ) / f ( x ) , x → + ∞ ; (1.14b)

which imply f ″ ( x ) ≠ 0 for almost all x large enough. The asymptotic relation in (1.14b) is more conveniently written as

f ″ ( x ) / f ( x ) ∼ ( f ′ ( x ) / f ( x ) ) 2 ,   x → + ∞ . (1.14c)

(V) (Rapid variation of higher order). A function f ∈ A C n [ T , + ∞ ) is called “rapidly varying at + ∞ of order n ≥ 2 (in the strong restricted sense)” if all the functions f , f ′ , ⋯ , f ( n − 1 ) are rapidly varying at + ∞ in the above-specified sense and this amounts to say that the following conditions hold true as x → + ∞ :

f ( k ) ( x ) ≠ 0     ∀   x   large   enough   and     0 ≤ k ≤ n ; (1.15)

f ( x ) / f ′ ( x ) = o ( x ) ;   f ′ ( x ) / f ″ ( x ) = o ( x ) ;   ⋯ ; f ( n − 1 ) ( x ) / f ( n ) ( x ) = o ( x ) ; (1.16)

( ( f ( x ) / f ′ ( x ) ) ′ = o ( 1 ) ;   ( f ′ ( x ) / f ″ ( x ) ) ′ = o ( 1 ) ;   ⋯ ; ( f ( n − 1 ) ( x ) / f ( n ) ( x ) ) ′ = o ( 1 ) ; (1.17)

wherein relations in (1.17) obviously imply those in (1.16).

If f is rapidly varying at + ∞ of order n ≥ 2 in the previous sense then all the functions f , f ′ , ⋯ , f ( n − 1 ) belong to the same class, either R − ∞ ( + ∞ ) or R + ∞ ( + ∞ ) , hence we shall use notation f ∈ { R ± ∞ ( + ∞ )   of   order   n } to denote that f enjoys the properties in (1.15)-(1.16)-(1.17) plus the corresponding value ± ∞ of the limit in (1.5). For an f ∈ A C n [ T , + ∞ ) satisfying (1.15) we have the following characterizations:

<< Relation

f ∈ { R + ∞ ( + ∞ )   of   order   n } ∪ { R − ∞ ( + ∞ )   of   order   n }

holds true if and only if the following equivalent sets of conditions are satisfied:

( f ′ ( x ) / f ( x ) ∼ f ″ ( x ) / f ′ ( x ) ∼ ⋯ ∼ f ( n ) ( x ) / f ( n − 1 ) ( x ) ∼ f ( n + 1 ) ( x ) / f ( n ) ( x ) , i . e .     D l ( f ( k ) ( x ) ) ∼ D l ( f ( x ) ) ,   x → + ∞ ,   1 ≤ k ≤ n ;

(1.18)

f ( k + 2 ) ( x ) ∼ ( f ( k + 1 ) ( x ) ) 2 / f ( k ) ( x ) ,   x → + ∞ ,   0 ≤ k ≤ n − 1 ; (1.19)

( f ( k + 2 ) ( x ) / f ( x ) ∼ ( f ′ ( x ) / f ( x ) ) k + 2 ≡ ( D l ( f ( x ) ) ) k + 2 ,   x → + ∞ , 0 ≤ k ≤ n − 1. (1.20)

It follows that even f ( n + 1 ) ( x ) ≠ 0 for almost all x large enough. >>

Remarks. (I) The concepts of regular or smooth variation of order n involve derivatives up to order n, whereas our restricted concept of rapid variation of order n involves derivatives up to order n + 1 .

(II) In the classical definitions of regular or rapid variation, even in the weak Karamata’sense, f is assumed ultimately strictly positive whereas in our definition of higher-order variation f is allowed to be either >0 or <0, the essential point being that it ultimately assumes only one strict sign. The above-added locution in parenthesis “(in the strong sense)” is meant to distinguish our theories from the classical ones wherein the limits in (1.5) are replaced by weaker asymptotic functional relations.

(III) To be consistent with the classical theory, notation “ f ∈ R α ( + ∞ ) ” is used for an (ultimately) strictly positive function whereas, when specifying the indexes of higher-order variation of a function f and its derivatives, we write “ | f ( k ) | ∈ R α ( + ∞ ) ” with the absolute values.

With the following notation for the iterated natural logarithms:

( l k ( x ) : = log ( log ( ⋯ ( log ︸ k x ) ⋯ ) ) ,   k ≥ 1 ,   ( defined   for   x   large   enough ) ; l 0 ( x ) : = x ; (1.21)

we have that:

- Typical functions in the class { R α ( + ∞ )   of   any   order   n ∈ ℕ } ,   α ∈ ℝ are:

( x α ⋅ [ ∏ k = 1 p 1 ( l k ( x ) ) β k ] ⋅ [ ∏ k = 1 p 2 exp ( c k ( log x ) γ k ) ] ⋅ [ ∏ k = 2 p 3 exp ( d k ( l k ( x ) ) δ k ) ] , α , β k , c k , d k ∈ ℝ ;   0 < γ k < 1 ;   0 < δ k ; (1.22)

provided that they do not reduce to a power x p , p ∈ ℕ , which belongs to the class { R p ( + ∞ )   of   exact   order   p + 1 } .

- Typical functions in the classes { R ± ∞ ( + ∞ )   of   any   order   n ∈ ℕ } are:

( R α ( x ) ⋅ [ ∏ k = 1 p 2 exp ( c k ( log x ) γ k ) ] ⋅ [ ∏ k = 2 p 3 exp ( d k x δ k ) ] , R α ∈ { R α ( + ∞ )   of   any   order   n ∈ ℕ } ; c k , d k ∈ ℝ ;   γ k > 1 ;   0 < δ k ; (1.23)

where the pertinent class, either { R − ∞ ( + ∞ ) } or { R + ∞ ( + ∞ ) } is determined by the behavior of the function as x → + ∞ , according as it converges to zero or diverges to ± ∞ .

- Each of the following functions

x α ;     x α + e − x ;     x α + e − x sin x     ( α ∈ ℝ ) ; (1.24)

belongs to the class “ { S R α ( + ∞ )   of   any   order   n ∈ ℕ } ” for any α ∈ ℝ . But, for α ∈ ℕ ∪ { 0 } they belong to the class { R α ( + ∞ )   of   exact   order   n + 1 } .

Here is a brief summary of the results. In §2 we study to what extent the elementary factorization

f ∈ R α ( + ∞ ) ,   α ≠ 0,   ⇔   { f ( x ) ≡ x α L ( x ) , L ∈ R 0 ( + ∞ ) } ,

has an analogue for higher-order variation. After remarking that such an analogue does indeed exist for higher-order smoothly-varying functions and that the mere inference from right to left holds true for higher-order regularly-varying functions under certain restrictions on the values of α , we prove that no such restrictions are needed for an important class of functions including those in (1.22). This very class of higher-order regularly-varying functions does not require any restrictions on the indexes when performing the operation of multiplication. In §3 we obtain non-obvious results on the types of higher-order variation of linear combinations (of arbitrary signs) of various functions. In §4 we list a number of combinatorial identities elementarily inferred from the formulas for higher derivatives of composite or inverse functions and show how these identities may simplify some proofs of previous results about operations with higher-order types of variation. Finally, in §5 there is a discussion about the order of regular variation of the inverse of a higher-order regularly-varying function, pointing out significant difficulties for certain exceptional values of the indexes. Apart from smooth variation, results involving other types of asymptotic variations are obtained via direct analytic computations and estimates of the higher-order derivatives as in [ [2]: §7].

Applications of the mentioned results to determining the asymptotic behaviors of certain Hankel determinants are currently being developed by the author, whereas future applications of the whole theory of higher-order types of asymptotic variation to ordinary or partial differential equations are hoped to be studied by the present or other authors.

As we know from [ [2]: §7.2] the product of two or more regularly-varying functions of order n ≥ 2 is again of order n only under some restrictions on the indexes of variation, restrictions quite unnatural in applications; so it is useful to point out some classes of functions wherein such restrictions are not needed, above all because these classes include most of the functions of interest in asymptotic questions as those in (1.22). A related question concerns the factorization of a higher-order regularly-varying function which we are going to treat first.

The exact evaluation of the index of variation of a linear combination of two functions is in general possible only under some restrictions: see [ [1]: Prop. 2.1-(iii), p. 784, claims in (2.27) and in the subsequent line] for regular variation, and [ [2]: relations in (7.4), p. 820] for higher-order smooth variation. For linear combinations of more than two functions we gave a result in [ [1]: Prop. 2.3-(II), p. 789], under “the least possible restrictions”, including regular or rapid variation. Extensions to more than two functions with higher-order regular or rapid variation require specifications in the proofs or in the statement. We give some extensions needed in applications.

In three different places in [2], we needed to show that certain constants were non-zero and either hinted at or provided indirect proofs. As a matter of fact, the exact values of the constants are special elementary cases of the classical formulas for the higher derivatives of composition and inversion, formulas extensively used in [2] and reported here from [ [2]: §6] together with some special cases. The proofs are so brief and elementary that we report them in full.

- Faà Di Bruno’s formula for higher derivatives of a composition:

( f ( g ( x ) ) ) ( k ) ≡ d k d x k ( f ( g ( x ) ) ) = ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k k ! i 1 !   ⋯ i k !   ( 1 ! ) i 1 ( 2 ! ) i 2 ⋯ ( k ! ) i k     × f ( i 1 + ⋯ + i k ) ( g ( x ) ) ⋅ ( g ′ ( x ) ) i 1 ⋅ ( g ″ ( x ) ) i 2 ⋯ ( g ( k ) ( x ) ) i k ,   k ≥ 1 , (4.1)

where the summation is taken over all possible ordered k-tuples of non-negative integers i j such that

i 1 + 2 i 2 + ⋯ + k i k = k     ( hence     1 ≤ i 1 + i 2 + ⋯ + i k ≤ k ) . (4.2)

In the preceding sum there is only one term containing f ( k ) and only one term containing g ( k ) , both with coefficient 1, namely:

( f ( k ) ( g ( x ) ) ⋅ ( g ′ ( x ) ) k corresponding   to   ( i 1 , i 2 , ⋯ , i k ) = ( k ,0, ⋯ ,0 ) ; f ′ ( g ( x ) ) ⋅ g ( k ) ( x ) corresponding   to   ( i 1 , i 2 , ⋯ , i k ) = ( 0, ⋯ ,0,1 ) . (4.3)

For convenience the coefficients into the sum in (4.1) will be denoted in the sequel by the symbol a i 1 , i 2 , ⋯ , i k .

Lemma 4.1. (Special cases of Di Bruno’s formula).

(I) Choosing “ f ( y ) : = y β ,   g ( x ) : = x α ” yields the identity:

∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k a i 1 , i 2 , ⋯ , i k ⋅ α 1 _ ⋅ α 2 _ ⋯ α k _ ⋅ β i 1 + ⋯ + i k _ = ( α β ) k _     ∀   α , β ≠ 0 ,   k ∈ ℕ , (4.4)

which follows from:

D k x α β = ( α β ) k _ ⋅ x α β − k = D k ( ( x α ) β ) = ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k a i 1 , i 2 , ⋯ , i k ⋅ β i 1 + ⋯ + i k _ ⋅ ( x α ) β − ( i 1 + ⋯ + i k )     × α 1 _ ⋅ α 2 _ ⋯ α k _ ⋅ x ( α − 1 ) i 1 + ( α − 2 ) i 2 + ⋯ + ( α − k ) i k = x α β − k ⋅ ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k a i 1 , i 2 , ⋯ , i k ⋅ α 1 _ ⋅ α 2 _ ⋯ α k _ ⋅ β i 1 + ⋯ + i k _ . (4.5)

(II) Choosing “ f ( y ) : = y α ,   g ( x ) : = e x ” yields the identity:

∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k     a i 1 , i 2 , ⋯ , i k ⋅ α i 1 + ⋯ + i k _ = α k     ∀   α ≠ 0 ,   k ∈ ℕ , (4.6)

which follows from:

D k e α x = α k ⋅ e α x = D k ( e x ) α = ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k a i 1 , i 2 , ⋯ , i k ⋅ α i 1 + ⋯ + i k _ ⋅ ( e x ) α − ( i 1 + ⋯ + i k ) ⋅ e ( i 1 + ⋯ + i k ) x = e α x ⋅ ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k a i 1 , i 2 , ⋯ , i k ⋅ α i 1 + ⋯ + i k _ . (4.7)

(III) Choosing “ f ( y ) : = e y ,   g ( x ) : = log x ” yields the identity:

( ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k     a i 1 , i 2 , ⋯ , i k ⋅ ( − 1 ) k − i 1 − ⋯ − i k × ( 0 ! ) i 1 ( 1 ! ) i 2 ( 2 ! ) i 3 ⋯ ( ( k − 1 ) ! ) i k = 0   ∀   k ∈ ℕ ,   k ≥ 2 , (4.8)

which follows from:

D k exp ( log x ) ≡ for   k ≥ 2 0 = ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k     a i 1 , i 2 , ⋯ , i k ⋅ x 1 − i 1 − 2 i 2 − ⋯ − k i k     × ( − 1 ) ( 2 − 1 ) i 2 + ( 3 − 1 ) i 3 + ⋯ + ( k − 1 ) i k ⋅ + ( 0 ! ) i 1 ( 1 ! ) i 2 ( 2 ! ) i 3 ⋯ ( ( k − 1 ) ! ) i k = x 1 − k ⋅ ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k     a i 1 , i 2 , ⋯ , i k ⋅ ( − 1 ) k − i 1 − ⋯ − i k     × ( 0 ! ) i 1 ( 1 ! ) i 2 ( 2 ! ) i 3 ⋯ ( ( k − 1 ) ! ) i k . (4.9)

(IV) And choosing “ f ( y ) : = log y ,   g ( x ) : = exp x ” yields the identity:

∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k     a i 1 , i 2 , ⋯ , i k ⋅ ( − 1 ) i 1 + ⋯ + i k − 1 ⋅ ( i 1 + ⋯ + i k − 1 ) ! = 0     ∀   k ∈ ℕ ,   k ≥ 2. (4.10)

- Ostrowski’s formula for higher derivatives of an inverse function.

For the inverse function of a k-time differentiable f ( x ) with f ′ ( x ) ≠ 0 , the formula holds true:

d k d y k ( f − 1 ( y ) ) = [ f ′ ( f − 1 ( y ) ) ] 1 − 2 k ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯ ( − 1 ) k − i 1 − 1 ( 2 k − i 1 − 2 ) ! i 2 !   i 3 !   ⋯ i k !   ( 2 ! ) i 2 ( 3 ! ) i 3 ⋯ ( k ! ) i k     × [ f ′ ( f − 1 ( y ) ) ] i 1 ⋅ [ f ″ ( f − 1 ( y ) ) ] i 2 ⋯ [ f ( k ) ( f − 1 ( y ) ) ] i k ,   k ≥ 1 , (4.11)

where the summation is taken over all ordered k-tuples of non-negative integers i j such that

i 1 + ⋯ + i k = k − 1 ;   i 1 + 2 i 2 + ⋯ + k i k = 2 k − 2. (4.12)

For convenience the coefficients into the sum in (4.11) will be denoted in the sequel by the symbol c i 1 , i 2 , ⋯ , i k .

Lemma 4.2. (Special cases of Ostrowski’s formula). The non-negative indexes i j appearing in the various sums below are subject to the restrictions in (4.12).

(I) For “ f ( x ) : = e x ” we trivially get the identity:

∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯     c i 1 , ⋯ , i k = ( − 1 ) k − 1 ( k − 1 ) !     ∀   k ∈ ℕ . (4.13)

(II) For “ f ( x ) : = log x ” we have:

( f ( j ) ( x ) = ( − 1 ) j − 1 ( j − 1 ) !   x − j ;   f − 1 ( y ) ≡ e y ; f ( j ) ( f − 1 ( y ) ) = ( − 1 ) j − 1 ( j − 1 ) !   e − j y ,   j ≥ 1 ; (4.14)

and:

D k e y ≡ e y = e ( 2 k − 1 ) y ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯     c i 1 , ⋯ , i k ⋅ ( − 1 ) ( 2 − 1 ) i 2 + ⋯ + ( k − 1 ) i k                                   × ( ( 2 − 1 ) ! ) i 2 ( ( 3 − 1 ) ! ) i 3 ⋯ ( ( k − 1 ) ! ) i k e ( − i 1 − 2 i 2 − ⋯ − k i k ) y                   = e y ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯     c i 1 , ⋯ , i k ⋅ ( − 1 ) k − 1 ( 1 ! ) i 2 ⋯ ( ( k − 1 ) ! ) i k ; (4.15)

so that we get the identity:

∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯     c i 1 , ⋯ , i k ⋅ ( − 1 ) k − 1 ( 1 ! ) i 2 ( 2 ! ) i 3 ⋯ ( ( k − 1 ) ! ) i k = 1     ∀   k ∈ ℕ . (4.16)

(III) For “ f ( x ) : = x α ,   α ≠ 0 ” we get the identity:

∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯     c i 1 , ⋯ , i k ⋅ ( α 1 _ ) i 1 ⋅ ( α 2 _ ) i 2 ⋯ ( α k _ ) i k = α 2 k − 1 ⋅ ( 1 / α ) k _     ∀   k ∈ ℕ , (4.17)

which follows from:

D k y 1 / α = ( 1 / α ) k _ ⋅ y ( 1 / α ) − k = ( α y 1 − ( 1 / α ) ) 1 − 2 k ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯ c i 1 , ⋯ , i k     × ( α 1 _ ) i 1 ⋅ ( α 2 _ ) i 2 ⋯ ( α k _ ) i k ⋅ y ( 1 − ( 1 / α ) ) i 1 + ( 1 − ( 2 / α ) ) i 2 + ⋯ + ( 1 − ( k / α ) ) i k = α 1 − 2 k y ( 1 / α ) − k ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯     c i 1 , ⋯ , i k ⋅ ( α 1 _ ) i 1 ⋅ ( α 2 _ ) i 2 ⋯ ( α k _ ) i k . (4.18)

We shall now use some of the foregoing identities to shorten or clarify some proofs in [2].

Proofs of Proposition 7.6-(II) in [ [2]: pp. 827-829] and of Proposition (9.4)-(I) in [ [2]: p. 850]. Relation (7.65) in [ [2]: p. 829] must read:

h ( k ) ( x ) = ( h ′ ( x ) ) k ( h ( x ) ) k − 1 ⋅ [ c k , α + o ( 1 ) ] , (4.19)

with suitable constants c k , α different from the A k ’s erroneously written therein, and provided that the c k , α ’s are non-zero. Looking at the preceding formula (7.64) in [2] we see that

c k , α = α − k ⋅ ∑ 0 ≤ i j ≤ k i 1 + 2 i 2 + ⋯ + k i k = k a i 1 , i 2 , ⋯ , i k ⋅ α i 1 + ⋯ + i k _ = by   ( 4.6 ) 1 , (4.20)

and the proof of Proposition 7.6-(II) is over. In [ [2]: p. 829] we hinted at an indirect proof of “ c k , α = 1 ” based on the last remark in [ [1]: pp. 810-811] but, admittedly, the pertinent argument is not so immediate as the reader might be led to think by the mere mention in [1]. Out of fairness, we give here the whole reasoning. The full statement of the remark in question is:

( f ∈ A C n [ T , + ∞ ) ;     f ( k ) ( x ) ≠ 0     ∀   x   large   enough   and   0 ≤ k ≤ n ; f ( j ) ∈ R ± ∞ ( + ∞ )     for   some   j ,   0 ≤ j ≤ k ; f ( k ) ( x ) ⋅ f ( k + 2 ) ( x ) ⋅ ( f ( k + 1 ) ( x ) ) − 2 = c k + o ( 1 ) ,   → + ∞ , for   some   k ∈ { 0,1, ⋯ , n − 1 } ; } ⇒   c k = 1. (4.21)

Now, in the context of Proposition 7.6-(II) we may argue that if the constants c k , α are known to be non-zero for all values of k and a fixed value of α , then relation (7.65) in [2] implies (with the pertinent notations):

h ( k ) ( x ) ⋅ h ( k + 2 ) ( x ) ⋅ ( h ( k + 1 ) ( x ) ) − 2 = [ c k , α + o ( 1 ) ] [ c k + 2, α + o ( 1 ) ] [ c k + 1, α + o ( 1 ) ] − 2 ≡ d k , α + o ( 1 ) , (4.22)

with a suitable constant d k , α ≠ 0 . Already knowing that “ h ∈ R ± ∞ ( + ∞ ) ”, we get d k , α = 1 from (4.21). After the easy direct checking that “ c k , α = 1 for k = 2 , 3 , 4 ” we infer that:

1 = d 3 , α = c 3 , α ⋅ c 5 , α ⋅ ( c 4 , α ) − 2     and     c 5 , α = 1 ;

and so on we get c k , α = 1 for all values of k. Now we are in a position to prove that no c k , α is zero assuming, if possible, that:

c i , α = 1     for     i = 2 , ⋯ , k + 1     and     c k + 2 , α = 0.

Then (4.22) would imply

h ( k ) ( x ) ⋅ h ( k + 2 ) ( x ) ⋅ ( h ( k + 1 ) ( x ) ) − 2 = o ( 1 ) ,

which is equivalent to

( h ( k ) ( x ) / h ( k + 1 ) ( x ) ) ′ = 1 + o ( 1 ) ,

and which, in turn, implies

h ( k ) ( x ) / h ( k + 1 ) ( x ) ∼ x ,   x → + ∞ ,     i . e .     h ( k ) ∈ R 1 ( + ∞ ) ,

inconsistently with the property “ h ∈ R ± ∞ ( + ∞ ) ”.

As concerns the proof of Proposition (9.4)-(I) in [ [2]: p. 850], in the final part we came across the sum now reported on the right-hand side in (4.20) and we gave another indirect proof of the identity in (4.6) intermixing algebraic arguments and facts from the theory of higher-order exponential variation.

□

Proofs of Propositions 7.7-(II) and 7.7-(III) in [ [2]: pp. 830-831]. In the concluding part of Proposition 7.7-(II) in [ [2]: p. 831] relation (7.79) contains a constant C k which is defined by

C k : = ∑ 0 ≤ i j ≤ k − 1 ⋯ ⋯     c i 1 , ⋯ , i k (4.23)

which is non-zero by (4.13), and this fact concludes the proof.

Similarly, in the proof of Proposition 7.7-(III) in [ [2]: p. 831], relation (7.82) contains a constant again denoted by C k . This constant is obtained by replacing relations (7.80) in [2], for the derivatives f ( i ) , into (7.76) in [2] which is exactly the operation performed in the claims in Lemma 4.2. This new constant is the sum appearing in our relation (4.16) which equals 1, and this concludes the proof.

□

We present a last instance wherein a combinatorial identity plays a role, namely a direct proof of a part of Proposition 7.7 in [2].

Proposition 4.3. (Former Proposition 7.7-(I) in [2] ). If

f ∈ { R α ( + ∞ )   of   order   n } ,   α > 0 ,   ( 1 / α ) ≠ 1 , 2 , ⋯ , n − 2 , (4.24)

then the inverse function

f − 1 ∈ { R 1 / α ( + ∞ )   of   order   n } . (4.25)

Proof. To avoid a mix-up over the exponents we put f ˜ : = f − 1 and f ˜ ( k ) : = D k f ˜ . Relations in (1.7) imply:

f ( i ) ( f ˜ ( y ) ) = ( f ˜ ( y ) ) − i ⋅ y ⋅ [ α i _ + o ( 1 ) ] ,   1 ≤ i ≤ n , (4.26)

which, when replaced into (4.11), yield:

f ˜ ( k ) ( y ) = [ f ′ ( f ˜ ( y ) ) ] 1 − 2 k   × ∑ 0 ≤ i j ≤ k − 1 ⋯     c i 1 , ⋯ , i k [ f ′ ( f ˜ ( y ) ) ] i 1 ⋅ [ f ″ ( f ˜ ( y ) ) ] i 2 ⋯ [ f ( k ) ( f ˜ ( y ) ) ] i k = ( f ˜ ( y ) ) 2 k − 1 ⋅ y 1 − 2 k ⋅ [ α + o ( 1 ) ]   × ∑ 0 ≤ i j ≤ k − 1 ⋯ { c i 1 , ⋯ , i k ( f ˜ ( y ) ) − i 1 − 2 i 2 − ⋯ − k i k ⋅ y i 1 + ⋯ + i k ⋅ ∏ j = 1 k [ α i _ + o ( 1 ) ] i j } = ( f ˜ ( y ) ) 2 k − 1 + 2 − 2 k ⋅ y 1 − 2 k + k − 1 ⋅ [ α + o ( 1 ) ] ⋅ [ d k , α + o ( 1 ) ] = f ˜ ( y ) ⋅ y − k ⋅ [ α ⋅ d k , α + o ( 1 ) ] , (4.27)

provided that the constant

α ⋅ d k , α ≡ α ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ { c i 1 , ⋯ , i k ⋅ ∏ j = 1 k ( α i _ ) i j }

is non-zero as granted by (4.17) and the restrictions on α . □

Referring to Proposition 4.3 the trivial counterexample of f ( x ) : = x 1 / p ,   p ∈ ℕ shows the necessity of the restrictions on α . Condition “ α > 0 ” only grants that the inverse function is defined on some neighborhood of + ∞ . For order n = 1 any α works well. Spurred on by the results in Proposition 2.4 we tried to suppress the above restrictions on α for the subclass of regularly-varying functions involved in Propositon 2.2 but the situation presents inherent difficulties and only a complete result for α = 1 is given here together with a partial result for the remaining exceptional values of α . The difficulties for a general result are outlined in §6.

Theorem 5.1. (I) A special case of Proposition 2.2 states that:

( | L ( k ) | ∈ R − k ( + ∞ ) ,   0 ≤ k ≤ n − 1 ,   n ≥ 2 , f ( x ) : = x L ( x ) , } ⇒   ( f ∈ { R 1 ( + ∞ )   of   order   n } , | f ( k ) | ∈ R 1 − k ( + ∞ ) ,   0 ≤ k ≤ n − 1. (5.1)

Here we are obviously assuming n ≥ 2 and the assumptions in (5.1) imply that the derivatives L ( k ) , k ≤ n − 1 , never vanish on a neighborhood of + ∞ hence L is ultimately strictly monotonic. Analogously, the conclusions in (5.1) imply that f is ultimately strictly monotonic. If L is ultimately positive then the inverse function f ˜ is defined on a neighborhood of + ∞ . It will be presently shown that f ˜ belongs to the same class of f and its derivatives satisfy the same relations as those of | f ( k ) | in (5.1).

(II) The following is a partial result for the other exceptional values of α :

( | L ( k ) | ∈ R − k ( + ∞ ) ,   0 ≤ k ≤ n − 1,   n ≥ 2, f ( x ) : = x 1 / p L ( x ) ,   p ∈ ℕ ,   p ≥ 2, } ⇒   | f ˜ ( k ) | ∈ R p − k ( + ∞ ) ,   0 ≤ k ≤ min { n − 1, p } . (5.2)

Proof. We are using the notation f ˜ : = f − 1 as in the proof of Proposition 4.3. (I) The function L satisfies relations in (2.12) whereas relations in (2.7)-(2.9) take the form:

( f ′ ( x ) = L ( x ) + x L ′ ( x ) ∼ L ( x ) ≡ x − 1 f ( x ) ; f ″ ( x ) ∼ L ′ ( x ) = o ( x − 1 L ( x ) ) = o ( x − 1 f ′ ( x ) ) ; f ( k ) ( x ) ∼ ( − 1 ) k ( k − 2 ) !   x 2 − k L ′ ( x ) ,   k ≥ 2 ; (5.3)

whence, with the natural substitution x = f ˜ ( y ) :

f ˜ ′ ( y ) ≡ [ f ′ ( f ˜ ( y ) ) ] − 1 ∼ f ˜ ( y ) ⋅ [ f ( f ˜ ( y ) ) ] − 1 ≡ y − 1 f ˜ ( y )       i . e .     f ˜ ∈ R 1 ( + ∞ ) ; (5.4)

( f ˜ ″ ( y ) ≡ − ( f ˜ ′ ( y ) ) 3 ⋅ f ″ ( f ˜ ( y ) ) ; f ″ ( f ˜ ( y ) ) = ( 5.3 ) o ( ( f ˜ ( y ) ) − 1 ⋅ f ′ ( f ˜ ( y ) ) ) ≡ ( 5.4 ) o ( [ f ˜ ( y ) ⋅ f ˜ ′ ( y ) ] − 1 ) ; (5.5)

f ˜ ″ ( y ) / f ˜ ′ ( y ) ≡ − ( f ˜ ′ ( y ) ) 2 ⋅ f ″ ( f ˜ ( y ) ) = ( 5.5 ) o ( f ˜ ′ ( y ) / f ˜ ( y ) ) = ( 5.4 ) = o ( y − 1 ) ,   y → + ∞ ,     i . e .     f ˜ ′ ∈ R 0 ( + ∞ ) . (5.6)

For k ≥ 3 we use Ostrowski’s formula (4.11) highlighting the signs of the coefficients and we get as y → + ∞ :

f ˜ ( k ) ( y ) = ( f ˜ ′ ( y ) ) 2 k − 1 ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ { | c i 1 , ⋯ , i k | ( − 1 ) k − i 1 − 1 ( − 1 ) 2 i 2 + ⋯ + k i k ( 0 ! ) i 2 ( 1 ! ) i 3 ⋯ ( ( k − 2 ) ! ) i k     × ( f ˜ ( y ) ) ( 2 − 2 ) i 2 + ( 2 − 3 ) i 3 + ⋯ + ( 2 − k ) i k ⋅ ( L ′ ( f ˜ ( y ) ) ) i 1 ⋅ ( L ′ ( f ˜ ( y ) ) ) i 2 ⋯ ( L ′ ( f ˜ ( y ) ) ) i k ⋅ [ 1 + o ( 1 ) ] } = ( f ˜ ′ ( y ) ) 2 k − 1 ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ { | c i 1 , ⋯ , i k | ( − 1 ) 3 ( k − 1 ) ( ∏ j = 0 k − 2 ( j ! ) i j + 2 ) ( f ˜ ( y ) ) − i 1     × ( L ( f ˜ ( y ) ) ) i 1 ⋅ ( L ′ ( f ˜ ( y ) ) ) k − 1 − i 1 ⋅ [ 1 + o ( 1 ) ] } = ( − 1 ) 3 ( k − 1 ) ⋅ ( f ˜ ′ ( y ) ) 2 k − 1 ⋅ ( L ′ ( f ˜ ( y ) ) ) k − 1     × ∑ 0 ≤ i j ≤ k − 1 ⋯ { c ¯ i 1 , ⋯ , i k ( L ( f ˜ ( y ) ) ) i 1 ⋅ ( L ′ ( f ˜ ( y ) ) ⋅ f ˜ ( y ) ) − i 1 ⋅ [ 1 + o ( 1 ) ] } , (5.7)

with suitable positive coefficients c ¯ i 1 , ⋯ , i k . We now put

F ( y ) : = L ( f ˜ ( y ) ) / ( L ′ ( f ˜ ( y ) ) ⋅ f ˜ ( y ) ) , (5.8)

and notice that the assumptions in (5.1) imply

x L ′ ( x ) / L ( x ) = o ( 1 ) ,   x → + ∞ ;   L ′ ( x ) ≠ 0     ultimately ; (5.9)

whence F ( y ) → ± ∞ ,   y → + ∞ , and:

∑ 0 ≤ i j ≤ k − 1 ⋯ { c ¯ i 1 , ⋯ , i k ( F ( y ) ) i 1 ⋅ [ 1 + o ( 1 ) ] } ∼ c k ( F ( y ) ) max i 1 ,   y → + ∞ , (5.10)

where c k > 0 and the number “ max i 1 ” a priori depends only on k. Now, the exponents i j satisfy the two equations in (4.12) whence i 1 < k − 1 ; and for the choice i 1 = k − 2 the system in (4.12) reduces to

i 2 + ⋯ + i k = 1 ,     2 i 2 + ⋯ + k i k = k , (5.11)

which has the solution “ i 2 = ⋯ = i k − 1 = 0 ,   i k = 1 ”. It follows that “ max i 1 = k − 2 ” and:

f ˜ ( k ) ( y ) ∼ ( − 1 ) 3 ( k − 1 ) c k ⋅ ( f ˜ ′ ( y ) ) 2 k − 1 ⋅ ( L ′ ( f ˜ ( y ) ) ) k − 1 ⋅ ( F ( y ) ) k − 2 ; (5.12)

( f ˜ ( k + 1 ) ( y ) / f ˜ ( k ) ( y ) ∼ − c k + 1 c k ( f ˜ ′ ( y ) ) 2 ⋅ L ′ ( f ˜ ( y ) ) ⋅ F ( y ) = − c k + 1 c k ( f ˜ ′ ( y ) ) 2 ⋅ ( L ( f ˜ ( y ) ) / f ˜ ( y ) ) . (5.13)

The first relation in (5.3) gives L ( f ˜ ( y ) ) ∼ f ′ ( f ˜ ( y ) ) = ( f ˜ ′ ( y ) ) − 1 and we get:

f ˜ ( k + 1 ) ( y ) / f ˜ ( k ) ( y ) ∼ − c k + 1 c k f ˜ ′ ( y ) / f ˜ ( y ) ∼ ( 5.4 ) − c k + 1 c k y − 1 ,   y → + ∞ , (5.14)

which means that “ f ˜ ( k ) ∈ R α k ( + ∞ ) ” for suitable α k < 0 and 2 ≤ k ≤ n − 1 . By (5.4) and (5.6) “ f ˜ ∈ { R 1 ( + ∞ )   of   order   n } ” and α 1 = 0 , and we shall now repeatedly use the general result about the index of variation of a derivative, [ [1]: Prop. 2.6-(I), p. 796], to show that α k = 1 − k . For the index of f ˜ ″ we preliminarily remark that

lim y → + ∞ f ˜ ′ ( y ) = ( 5.4 ) lim y → + ∞ y − 1 f ˜ ( y ) = lim x → + ∞ ( f ( x ) ) − 1 ⋅ x ≡ lim x → + ∞ ( L ( x ) ) − 1 , (5.15)

and that “ lim x → + ∞ L ( x ) ” exists in [ 0, + ∞ ] by the positivity and monotonicity of L ( x ) .

First case. If this last limit is either zero or + ∞ then the same is true (with inverted values) for the “ lim y → + ∞ f ˜ ′ ( y ) ” and, by the mentioned general result, f ˜ ″ ∈ R − 1 ( + ∞ ) . This in turn implies the mentioned values of the indexes for the higher-order derivatives.

Second case. If “ lim x → + ∞ L ( x ) = l ∈   ] 0, + ∞ ) ” we may suppose l = 1 and represent f in the form “ f ( x ) : = x [ 1 + L ¯ ( x ) ] ” where the new function L ¯ satisfies the same assumptions satisfied by L in (5.1) and “ L ¯ ( x ) = o ( 1 ) ”. All the previous calculations from (5.3) to (5.14) remain valid because no use was made of the value L ( + ∞ ) , hence “ f ˜ ( k ) ∈ R α k ( + ∞ ) ” for suitable α k < 0 and 2 ≤ k ≤ n − 1 . In the present case we need to make explicit the asymptotic behavior of f ˜ ″ and we have:

( f ( x ) = x [ 1 + L ¯ ( x ) ] ; f ′ ( x ) = 1 + L ¯ ( x ) + x L ¯ ′ ( x ) = 1 + L ¯ ( x ) [ 1 + o ( 1 ) ] = 1 + o ( 1 ) ; f ″ ( x ) = 2 L ¯ ′ ( x ) + x L ¯ ″ ( x ) = 2 L ¯ ′ ( x ) − L ¯ ′ ( x ) [ 1 + o ( 1 ) ] = L ¯ ′ ( x ) [ 1 + o ( 1 ) ] ; (5.16)

( log y = log x + log [ 1 + L ¯ ( x ) ] = log x + o ( 1 ) ,   x → + ∞ ,     where     y ≡ f ( x ) ; log x = log y + o ( 1 ) ,   y → + ∞ ,     and     x ≡ f ˜ ( y ) = y [ 1 + o ( 1 ) ] ,   y → + ∞ ; (5.17)

whence:

( f ˜ ′ ( y ) ≡ [ f ′ ( f ˜ ( y ) ) ] − 1 = [ 1 + o ( 1 ) ] ; f ˜ ″ ( y ) ≡ − ( f ˜ ′ ( y ) ) 3 ⋅ f ″ ( f ˜ ( y ) ) ∼ − f ″ ( f ˜ ( y ) ) ∼ − L ¯ ′ ( f ˜ ( y ) )                   = − L ¯ ′ ( y [ 1 + o ( 1 ) ] ) ∼ − L ¯ ′ ( y ) ,   y → + ∞ ; (5.18)

where the last expression follows from a result about asymptotic functional relations: [ [1]: Prop. 5.1-(I), p. 811]. As “ L ¯ ′ ∈ R − 1 ( + ∞ ) ” by hypothesis, the relation “ f ˜ ″ ( y ) ∼ − L ¯ ′ ( y ) ” implies α 2 = − 1 by [ [1]: first inference in (2.31), p. 785]. The values of the other α k ’s follow automatically as pointed out in the first case and the proof of part (I) is over.

(II) Putting α : = 1 / p , relations in (2.7) state that:

f ( k ) ( x ) ∼ α k _ x α − k L ( x ) ,   x → + ∞ ,   1 ≤ k ≤ n − 1, (5.19)

so that:

f ˜ ′ ( y ) ≡ [ f ′ ( f ˜ ( y ) ) ] − 1 ∼ p ⋅ ( f ˜ ( y ) ) 1 − α ⋅ ( L ( f ˜ ( y ) ) ) − 1 ≡ p ⋅ f ˜ ( y ) ⋅ y − 1 ; (5.20)

f ˜ ( k ) ( y ) = ( f ˜ ′ ( y ) ) 2 k − 1 ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ { | c i 1 , ⋯ , i k | ( − 1 ) k − 1 − i 1 ( α 1 _ ) i 1 ( α 2 _ ) i 2 ⋯ ( α k _ ) i k     × ( f ˜ ( y ) ) ( α − 1 ) i 1 + ( α − 2 ) i 2 + ⋯ + ( α − k ) i k ⋅ ( L ( f ˜ ( y ) ) ) i 1 + i 2 + ⋯ + i k ⋅ [ 1 + o ( 1 ) ] } = ( f ˜ ′ ( y ) ) 2 k − 1 ⋅ ( f ˜ ( y ) ) ( α − 2 ) ( k − 1 ) ( L ( f ˜ ( y ) ) ) k − 1     × ( − 1 ) k − 1 ⋅ ∑ 0 ≤ i j ≤ k − 1 ⋯ { | c i 1 , ⋯ , i k | ( − 1 ) i 1 ( α 1 _ ) i 1 ( α 2 _ ) i 2 ⋯ ( α k _ ) i k ⋅ [ 1 + o ( 1 ) ] } . (5.21)

At this point the proof could be completed if we only knew that the sum

∑ 0 ≤ i j ≤ k − 1 ⋯ | c i 1 , ⋯ , i k | ( − 1 ) i 1 ( α 1 _ ) i 1 ( α 2 _ ) i 2 ⋯ ( α k _ ) i k (5.22)

is non-zero which may not be the case. As a matter of fact, for L ( x ) ≡ 1 we have f ˜ ( y ) ≡ y p and the above formula gives the expression of ( d k / d y k ) y p ≡ p k _ y p − 1 ; hence the sum in (5.22) equals ( − 1 ) k − 1 p k _ , being non-zero for k ≤ p and zero for k ≥ p + 1 . Hence the following relations hold true for the admissible values of k:

f ˜ ( k ) ( y ) ∼ c k y 1 − 2 k ( f ˜ ( y ) ) 2 k − 1 + ( α − 2 ) ( k − 1 ) ( L ( f ˜ ( y ) ) ) k − 1 ,   1 ≤ k ≤ p ;   c k ≠ 0 ; (5.23)

f ˜ ( k + 1 ) ( y ) / f ˜ ( k ) ( y ) ∼ c k + 1 c k y − 2 ( f ˜ ( y ) ) α ⋅ L ( f ˜ ( y ) ) ≡ c k + 1 c k y − 1 ,   1 ≤ k ≤ p − 1 ; (5.24)

where the last equality follows from the definition of f : ( f ˜ ( y ) ) 1 / p ⋅ L ( f ˜ ( y ) ) ≡ f ( f ˜ ( y ) ) . Hence f ˜ ∈ R α k ( + ∞ ) for certain indexes α k and, by [ [1]: Prop. 2.6-(I), p. 796], α k = p − k for k ≤ p − 1 . Moreover:

f ˜ ( p + 1 ) ( y ) = o ( y − 1 f ˜ ( p ) ( y ) )     hence     f ˜ ( p ) ∈ R 0 ( + ∞ ) . (5.25)

□

I) The main facts presented in this paper are:

1) The elementary factorization of regularly-varying functions:

f ∈ R α ( + ∞ ) ,   α ≠ 0,   ⇔   { f ( x ) ≡ x α L ( x ) , L ∈ R 0 ( + ∞ ) } , (6.1)

which holds true for higher-order smoothly-varying functions as well, cannot be extended to higher-order regularly-varying functions. It is known that the analogue of the sole inference from right to left in (6.1) holds true for higher-order regularly-varying functions under certain restrictions on α and we have shown in §2 that no such restrictions are needed for a useful class of functions including those in (1.22).

2) The foregoing result in turn implies that there is an important class of higher-order regularly-varying functions that require no impractical restrictions on their indexes of variation when performing on them the operation of multiplication, and this is proved in §2 as well.

3) Useful non-obvious results can be obtained on the types of higher-order variation for arbitrary linear combinations of various functions where “arbitrary” means “of any signs”, and appropriate counterexamples can be exhibited highlighting once more the special restricted character of our concept of higher-order rapid variation defined by relations in (1.15)-(1-17). The most meaningful results are those in Proposition 3.4 concerning linear combinations of both smoothly- and rapidly-varying functions. §3 ends with a non-trivial counterexample showing that the imposed assumptions are the least possible ones.

II) In §4 there is a list of combinatorial identities, directly inferred from the formulas for higher derivatives of composite or inverse functions, which are used to simplify some proofs of previous results about operations with higher-order types of variation. The reader may notice that the whole matter of operations with higher-order types of asymptotic variations, amply developed in [2], is quite complicated and some sporadic alternative proofs as those in §4, though longer, may be considered to provide support for these so many results.

III) In contrast to the results in §2 and §3 only partial results can be given in studying the type of higher-order variation of the inverse of a regularly-varying function and trying to suppress restrictions on the indexes. But this is due to inherent difficulties. It is the author’s firm conviction that the thesis in (5.2) holds true for all the admissible values of k ≥ p + 1 but, for derivatives of order greater than p it is not clear how to proceed by direct calculations in order to highlight the principal part of f ˜ ( k ) ( y ) and, consequently, of the ratio f ˜ ( k + 1 ) ( y ) / f ˜ ( k ) ( y ) . The difficulty is due to the factor ( − 1 ) i 1 in (5.22) which does not automatically assure that the constant is non-zero because the global sign of the other factors in (5.22) does not depend on the various exponents i j :

( α ≡ 1 / p ∈ ] 0 , 1 [   ⇒   sign   α j _ = ( − 1 ) j − 1   ⇒   sign   ( α j _ ) i j = ( − 1 ) ( j − 1 ) i j ⇒   sign   ∏ j = 1 k ( α j _ ) i j = ( − 1 ) ( 2 − 1 ) i 2 + ( 3 − 1 ) i 3 + ⋯ + ( k − 1 ) i k = ( − 1 ) k − 1 . (6.2)

The factor ( − 1 ) i 1 will be always present even if a two-term expansion of f ( k ) ( x ) , inferred from (2.13), is used, namely:

f ( k ) ( x ) = α k _ x α − k L ( x ) + x α − k + 1 L ′ ( x ) [ A k ( α ) + o ( 1 ) ] , (6.3)

noticing that even the coefficient A k ( α ) may be zero as, e.g., A 2 ( 1 / 2 ) = 0 .

We tried some calculations for a function f of the type appearing in (1.22) with α > 0 and not reduced to power. In this special case, each derivative f ( k ) is a linear combination of functions with the same algebraic structure of f (i.e. powers times the sort of slowly-varying functions specified therein) and, as such, it is either ≡ 0 , a case excluded by the assumption that f is not a power, or has a principal part as x → + ∞ with the same algebraic structure: hence f ( k ) ( x ) either converges to zero or diverges to ± ∞ as x → + ∞ . This grants that the limit “ lim x → + ∞ x f ( k + 1 ) ( x ) / f ( k ) ( x ) ” exists as an extended real number for each k ≥ 0 and, by the cited reference [ [1]: Prop. 2.6-(I), p. 796], “ | f ( k ) | ∈ R α − k ( + ∞ )   ∀   k ∈ ℕ ”.

Unfortunately, it is not obvious how to proceed to show the analogous property for the inverse function, i.e. that the limit “ lim y → + ∞ y f ˜ ( k + 1 ) ( y ) / f ˜ ( k ) ( y ) ” exists as an extended real number for each k ∈ ℕ implying that “ f ˜ ∈ { R p ( + ∞ )   of   any   order   n } ”.

This remains an “Open Problem” in addition to those stated in [ [2]: p. 866] save for Open Problem 4 completely solved in [3].

The author declares no conflicts of interest regarding the publication of this paper.

Granata, A. (2021) Operations with Higher-Order Types of Asymptotic Variation: Filling Some Gaps. Advances in Pure Mathematics, 11, 687-716. https://doi.org/10.4236/apm.2021.118046