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Complements to the Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions

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ABSTRACT

The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series; 2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians; 3) a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation; 4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.

1. Introduction

This paper is a direct continuation of [1] [2] and contains some complements to the theory developed therein with the purpose of completing the general theory. This first section, besides a list of notations, contains a summary of the various involved classes of functions and their main characterizations.

­ §2 contains a detailed study of the possible types of asymptotic variation of a convergent infinite series n = 1 c n ϕ n ( x ) where { ϕ n ( x ) } n is an asymptotic sequence at + and each { ϕ n ( x ) } has a definite index of variation. Two quite different situations occur depending on whether

ϕ 1 ( x ) ϕ n ( x ) , x + , or ϕ 1 ( x ) ϕ n ( x ) , x + .

The first case is elementary whereas, in treating the second case, we give non-trivial extensions of one known result for power series.

­ §3 contains results concerning the type of asymptotic variation of a Wronskian whose arguments are functions with a definite index of asymptotic variation at + so completing the extensive study of the asymptotic behaviors of Wronskians developed in two previous papers. The obtained results are quite natural and are based on the asymptotic study of a Vandermonde determinant with a gap in the exponents, a study which parallels the analogous investigation for standard Vandermondians in the previous papers.

­ §4 contains a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation. The theory developed in [1] [2] is necessarily based on asymptotic differential equations in order to define higher-order types of variation for differentiable functions, whereas the more general Karamata theory is based on asymptotic functional equations. We show that for a function with a monotonic derivative the two approaches coincide for each one of the studied classes of functions, a result already known for regular variation.

­ §5 contains a discussion about the concept of logarithmic variation starting from suitable asymptotic functional equations and showing, as in the previous section, the equivalence with corresponding asymptotic differential equations. Some of the results may be found in the literature but hidden in a quite-complicated general theory. The studied concept (namely, three related concepts) completes the list of the fundamental types of asymptotic variation in the way that we wished to systematize this theory.

­ §6 gives results on the inverse of a function with a definite type of exponential variation. The results require some calculations and are clarified by the concepts of logarithmic variation.

­ §7 contains some minor complements to the theory.

­ §8 contains the conclusions about the whole theory developed in three papers.

­ §9 contains a few bibliographical notes and a list of corrections for [1] [2] .

Here is a list of general notations used in [1] [2] .

­ : = { 1 , 2 , } ; : = { 0 , ± 1 , ± 2 , } ;

­ : = realline ; ¯ extendedrealline : = { ± } ;

­ 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 relation:

f ( x ) = + ( g ( x ) ) , x x 0 ( x I ) def ( f ( x ) = h ( x ) g ( x ) x near x 0 , lim x x 0 , x I h ( x ) = + ; (1.2)

and a similar definition for notation f ( x ) = ( g ( x ) ) , x x 0 ( x I ) . In particular:

f ( x ) = ± ( 1 ) , x x 0 ( x I ) def lim 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 , k ; (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.

­ Notation for the iterated natural logarithm:

( l k ( x ) : = log ( log ( ( log k x ) ) ) , k 1, ( defined for x large enough ) ; l 0 ( x ) : = x ; d d x ( l k ( x ) ) = ( i = 0 k 1 l i ( x ) ) 1 , k 1. (1.5)

For the reader’s convenience we give a list of the special classes of functions characterized in [1] [2] mentioning only the main facts to be used in the present paper.

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):

lim x + x f ( x ) / f ( x ) = { 0 ( slow variation at + ) , α \ { 0 } ( regular variation at + ) , ± ( rapid variation at + ) . (1.6)

(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 . (1.7)

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 , (1.8)

hold true whichever α 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 ; (1.9)

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 (1.8), i.e. f ( n ) , may have an arbitrary sign if α n _ = 0 .

(III) (Smooth variation). Relations in (1.8) characterize higher-order regular variation only for α { 0,1, , n 2 } ; as these relations are basic in the applications their validity defines 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 (1.8), 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 ; (1.10)

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

x k f ( k ) ( x ) / f ( x ) = α ( α 1 ) ( α k + 1 ) + o ( 1 ) , x + , 1 k n ; (1.11)

( x f ( x ) / f ( x ) = α + o ( 1 ) , x + ; ( x f ( x ) / f ( x ) ) ( k ) = o ( x k ) , x + , 1 k n . (1.12)

(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 + ; (1.13a)

or, equivalently, if:

( f ( x ) , f ( x ) 0 x large enough ; f ( x ) / f ( x ) f ( x ) / f ( x ) , x + ; (1.13b)

which imply f ( x ) 0 for almost all x large enough.

(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 ; (1.14)

f ( x ) / f ( x ) = o ( x ) ; f ( x ) / f ( x ) = o ( x ) ; ; f ( n 1 ) ( x ) / f ( n ) ( x ) = o ( x ) ; (1.15)

( f ( x ) / f ( x ) ) = o ( 1 ) ; ( f ( x ) / f ( x ) ) = o ( 1 ) ; ; ( f ( n 1 ) ( x ) / f ( n ) ( x ) ) = o ( 1 ) . (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 (1.14)-(1.15)-(1.16) plus the corresponding value ± of the limit in (1.6). For an f A C n [ T , + ) satisfying (1.14) we have the characterizations that conditions in (1.16) hold true, i.e.

f { R + ( + ) of order n } { R ( + ) of order n } ,

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 ; (1.17)

f ( k + 2 ) ( x ) ( f ( k + 1 ) ( x ) ) 2 / f ( k ) ( x ) , x + , 0 k n 1 ; (1.18)

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

(V) (Types of exponential variation). If f A C n 1 [ T , + ) then:

f { E 0 ( + ) of order n } ( f ( k ) ( x ) 0 x large enough and 0 k n 1, f ( x ) f ( x ) f ( n 1 ) ( x ) f ( n ) ( x ) , x + ; (1.19)

f { E c ( + ) of order n } , c \ { 0 } ( f ( x ) 0 x large enough , f ( k ) ( x ) c k f ( x ) , x + , 1 k n ; (1.20)

f { E ± ( + ) of order n } ( f ( x ) 0 x large enough , f ( n ) ( x ) f ( n 1 ) ( x ) f ( x ) f ( x ) , x + , (1.21)

wherein the correct index “ + ” or “ ” is determined by the single limit “ lim x + f ( x ) / f ( x ) ”.

For c = 0 there is no sign-restriction on the highest-order derivative f ( n ) , whereas for c 0 also f ( n ) turns out to be ultimately of one strict sign. More precisely, if f ( x ) > 0 and f { E c ( + ) of order n } then:

( c < 0 lim x + ( 1 ) k f ( k ) ( x ) = + , 0 k n ; 0 < c + lim x + f ( k ) ( x ) = + , 0 k n . (1.22)

Notice that 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 reader must remember that this is a semiexpository paper like [1] [2] and, as such, some elementary or known facts are explicitly reported or proved to have an exposition self-contained and easily-read.

2. Types of Asymptotic Variation of Infinite Series

In [1] and [2] there are some results about the index of variation of a linear combination of functions belonging to one of the previously-studied classes; in this section we give some results about the type of asymptotic variation of an infinite series of such functions. We know from ( [1]; formula (2.27), p. 784) that

{ ϕ i R α i ( + ) , α i , c i > 0 , 1 i n } i = 1 n c i ϕ i R max α i ( + ) , (2.1)

and similar results, with some restrictions, hold true when rapid variation is involved as stated in ( [1]; Prop. 2.3-(I), pp.788-789). Hence, when investigating the possible types of asymptotic variation of an infinite linear combination i = 1 c i ϕ i ( x ) , c i > 0 , there must be an essential difference between the two circumstances

ϕ 1 ( x ) ϕ n ( x ) , x + , (2.2)

ϕ 1 ( x ) ϕ n ( x ) , x + . (2.3)

The simplest case is (2.2) and here are two elementary results extending Proposition 2.3-(II) in ( [1]; p. 789), with no restriction on the signs of the coefficients c i . The C 1-regularity assumption simplifies the exposition.

Proposition 2.1. Let the functions ϕ n C 1 [ T , + ) , n , form the asymptotic scale (2.2), let ϕ 1 ( x ) 0 x and let { c n } n be a given sequence of arbitrary real numbers with c 1 0 .

(I) Assume the following further conditions:

| ϕ n ( x ) | A n | ϕ 1 ( x ) | q ( x ) x T n 2 ; n = 1 | c n | A n < + ; (2.4)

| ϕ n ( x ) | B n | ϕ 1 ( x ) | r ( x ) x T n 2 ; n = 1 | c n | B n < + ; (2.5)

where both q , r are suitable nonnegative functions such that

q , r bounded on [ T , + ) and = o ( 1 ) , x + . (2.6)

Then the two series

n = 1 c n ϕ n ( x ) , n = 1 c n ϕ n ( x ) (2.7)

are absolutely and uniformly convergent on each bounded interval of [ T , + ) . If f ( x ) is the sum of the first series then f C 1 [ T , + ) , f ( x ) = n = 1 c n ϕ n ( x ) and

f ( x ) / f ( x ) ϕ 1 ( x ) / ϕ 1 ( x ) , x + . (2.8)

(II) Assume all the conditions in part (I) with the exception that (2.5) is now replaced by

| ϕ n ( x ) / ϕ n ( x ) | C | ϕ 1 ( x ) / ϕ 1 ( x ) | x T n 2, (2.9)

or, more generally, by

| ϕ n ( x ) / ϕ n ( x ) | C n | ϕ 1 ( x ) / ϕ 1 ( x ) | x T n 2 , n = 1 | c n | A n C n < + . (2.10)

Then all the conclusions in part (I) still hold true. In the special case

ϕ n R α n ( + ) , α 1 > > α n > , ( even α 1 = + ) , (2.11)

relations in (2.2) are automatically satisfied ( [1]; Prop. 2.3-(III), p. 789), and the sequence { ϕ n ( x ) } n is an asymptotic scale as well provided that “ α n 0 n ” because of relations “ ϕ n ( x ) α n x 1 ϕ n ( x ) ”.

Remarks. Conditions in (2.4), (2.5), (2.10) are a kind of uniformity respectively for the infinite families of asymptotic relations: c n ϕ n ( x ) = o ( ϕ 1 ( x ) ) , c n ϕ n ( x ) = o ( ϕ 1 ( x ) ) , c n ϕ n ( x ) / ϕ n ( x ) = O ( ϕ 1 ( x ) / ϕ 1 ( x ) ) , n 2 .

These conditions cannot be dispensed with and even in a simple case such as

ϕ n R α n ( + ) ; α 1 > α 2 α n ; α n \ { 0 } , (2.12a)

the asymptotic relations

ϕ n ( x ) / ϕ n ( x ) ( α n / α 1 ) ( ϕ 1 ( x ) / ϕ 1 ( x ) ) , x + , (2.12b)

do not in themselves grant (2.10) as shown by the counterexample of

ϕ n ( x ) : = x 1 / n ( log x ) n , x 1 ; ϕ n ( x ) / ϕ n ( x ) = ( 1 n + n log x ) ( ϕ 1 ( x ) / ϕ 1 ( x ) ) . (2.13)

Proof. For part (I) the estimates

n = k | c n ϕ n ( x ) | ( n = k | c n | A n ) | ϕ 1 ( x ) | q ( x ) = o ( ϕ 1 ( x ) ) , k 2 ; (2.14)

n = k | c n ϕ n ( x ) | ( n = k | c n | B n ) | ϕ 1 ( x ) | r ( x ) = o ( ϕ 1 ( x ) ) , k 2 ; (2.15)

imply the assertions concerning the convergence of the two series due to the local boundedness of ϕ 1 , ϕ 1 , q , r , and moreover:

f ( x ) = c 1 ϕ 1 ( x ) + n = 2 c n ϕ n ( x ) c 1 ϕ 1 ( x ) ; f ( x ) = c 1 ϕ 1 ( x ) + n = 2 c n ϕ n ( x ) c 1 ϕ 1 ( x ) . (2.16)

For part (II) instead of (2.15) we now have:

n = k | c n ϕ n ( x ) | = n = k | c n ϕ n ( x ) | | ϕ n ( x ) ϕ n ( x ) | | ϕ 1 ( x ) ϕ 1 ( x ) | n = k C n | c n ϕ n ( x ) | ( n = k | c n | A n C n ) | ϕ 1 ( x ) | q ( x ) ; (2.17)

for k 2 and the subsequent conclusions are still valid.

An elementary example. For any function ϕ such that:

ϕ C 1 [ T , + ) ; ϕ ( x ) = o ( 1 ) , x + ; 0 < ϕ ( x ) M x T ; (2.18)

we have the estimates:

ϕ n ( x ) : = ( ϕ ( x ) ) n = ( ϕ ( x ) ) n 1 ϕ ( x ) M n 1 ϕ ( x ) , n 1 ; | ϕ n ( x ) / ϕ n ( x ) | = n | ϕ ( x ) / ϕ ( x ) | , n 2. (2.19)

Whatever M > 0 and k :

{ f k ( x ) : = n k 1 n ! ( ϕ ( x ) ) n = exp ( ϕ ( x ) ) 1 ϕ ( x ) 1 2 ! ( ϕ ( x ) ) 2 1 ( k 1 ) ! ( ϕ ( x ) ) k 1 ~ 1 k ! ( ϕ ( x ) ) k , x + ; f k ( x ) = n k 1 ( n 1 ) ! ( ϕ ( x ) ) n 1 ϕ ( x ) ~ 1 ( k 1 ) ! ( ϕ ( x ) ) k 1 ϕ ( x ) , x + . (2.20)

And if 0 < M < 1 and k :

{ g k ( x ) : = n k ( ϕ ( x ) ) n = 1 1 ϕ ( x ) 1 ϕ ( x ) ( ϕ ( x ) ) 2 ( ϕ ( x ) ) k 1 ~ ( ϕ ( x ) ) k , x + ; g k ( x ) = n k n ( ϕ ( x ) ) n 1 ϕ ( x ) ~ k ( ϕ ( x ) ) k 1 ϕ ( x ) , x + . (2.21)

In both cases “ f k ( x ) / f k ( x ) ~ k ϕ k ( x ) / ϕ k ( x ) ” and “ g k ( x ) / g k ( x ) ~ k ϕ k ( x ) / ϕ k ( x ) ”, hence:

ϕ R α ( + ) , α 0 f k , g k R k α ( + ) , ( k ) , (2.22)

the circumstance “ α > 0 ” being inconsistent with condition “ ϕ = o ( 1 ) ”.

A less elementary example. Consider the sequence of “modified iterated logarithms”:

( l ¯ 1 ( x ) : = log x , l ¯ n ( x ) : = log ( 1 + l ¯ n 1 ( x ) ) , n 2 ; l ¯ n C [ 1 , + ) ; d d x l ¯ n ( x ) = [ x ( 1 + l ¯ n 1 ( x ) ) ( 1 + l ¯ n 2 ( x ) ) ( 1 + l ¯ 1 ( x ) ) ] 1 , n 2 ; (2.23)

which satisfy “ l ¯ n ( x ) , d d x l ¯ n ( x ) > 0 for x > 1 ” and:

l ¯ 1 ( x ) l ¯ 2 ( x ) l ¯ n ( x ) ; d d x l ¯ 1 ( x ) d d x l ¯ 2 ( x ) d d x l ¯ n ( x ) , x + . (2.24)

To get useful estimates for our aim we must take x x 0 > 1 ; for instance, using the elementary inequality “ log ( 1 + x ) < x for x > 0 ” we get the following estimates for x e :

{ l ¯ 2 ( x ) l 2 ( x ) + log ( 1 + ( log x ) 1 ) < l 2 ( x ) + ( log x ) 1 , x e ; l ¯ 3 ( x ) l ¯ 2 ( x ) < l 2 ( x ) + ( log x ) 1 , x e ; l ¯ n ( x ) < l 2 ( x ) + ( log x ) 1 l 2 ( x ) + 1 , x e , n 2 ; (2.25)

whence:

{ 0 < l ¯ n ( x ) / l ¯ 1 ( x ) [ l 2 ( x ) + 1 ] ( log x ) 1 q ( x ) , ( A n 1 ) , x e , n 2 ; 0 < d d x l ¯ n ( x ) / d d x l ¯ 1 ( x ) = [ ( 1 + l ¯ n 1 ( x ) ) ( 1 + l ¯ n 2 ( x ) ) ( 1 + l ¯ 1 ( x ) ) ] 1 ( 1 + l ¯ 1 ( x ) ) 1 r ( x ) , ( B n 1 ) , x e , n 2. (2.26)

It follows that for any sequence { c n } n such that n 1 | c n | < + , both series

f ( x ) : = n = 1 c n l ¯ n ( x ) , f ( x ) : = n = 1 c n d d x l ¯ n ( x ) (2.27)

are absolutely convergent on [ e , + ) and uniformly convergent on each bounded interval and f ( x ) / f ( x ) d d x l ¯ 1 ( x ) / l ¯ 1 ( x ) , x + , i.e. f is slowly varying.

Let us now examine the case (2.3), the classical case being that of a power series with an infinite radius of convergence; here coefficients of nonconstant signs may generate entire functions with no definite type of asymptotic variation at + such as the trigonometric functions, hence in this case we must restrict our study to positive coefficients. A problem solved in American Mathematical Monthly, [3] , states that if the function f ( x ) : = n = 0 c n x n , with c n 0 , is defined for all x and k and is not a polynomial then f R + ( + ) , a result that can be extended to infinite series of regularly-varying functions.

Proposition 2.2. Assumptions:

ϕ n C 1 [ T , + ) and ϕ n R α n ( + ) , α n \ { 0 } , n ; (2.28)

ϕ n ( x ) > 0 , ϕ n ( x ) 0 , c n > 0 x [ T , + ) n ; (2.29)

f ( x ) : = n = 1 c n ϕ n ( x ) and f ( x ) = n = 1 c n ϕ n ( x ) uniformly convergent on each compact interval ; (2.30)

x ϕ n ( x ) = α n ϕ n ( x ) [ 1 + A n ( x ) ] with ( | A n ( x ) | A ( x ) = o ( 1 ) , x + α n 0 , n ; (2.31)

n = 1 c n | α n | ϕ n ( x ) convergent on each compact interval . (2.32)

We also assume (2.3) which we express by saying that { ϕ n ( x ) } n forms an “inverted asymptotic scale at + ”; this implies “ { α n } n nondecreasing” and we put α : = lim n α n ; hence α n α n . We separate the cases: “ < α < 0 ; α = 0 ; 0 < α + ”. In the first two cases monotonicity and restriction α n 0 imply “ α n < 0 n ”; whereas in the third case we may suppose, without loss of generality, that “ α n > 0 n ” due to the fact that for the sum of a finite number of terms we have n = 1 p ϕ n ( x ) R p ( + ) and we then split the given series “ n = 1 = n = 1 p + n > p ” and apply the results below. These agreements on the signs of α n imply that:

| n = 1 c n x ϕ n ( x ) | = n = 1 c n x | ϕ n ( x ) | ; | n = 1 c n α n ϕ n ( x ) | = n = 1 c n | α n | ϕ n ( x ) . (2.33)

Thesis:

x f ( x ) = ( n = 1 c n α n ϕ n ( x ) ) [ 1 + r ( x ) ] , with | r ( x ) | A ( x ) ; (2.34)

f R α ( + ) whichever the value of α : < α + . (2.35)

Remark. Of course, instead of (2.30) it is enough to assume that c n ϕ n ( x ) is uniformly convergent on each compact interval and that c n ϕ n ( x ) converges for at least one value of x; but in the present context it may sometimes be more convenient to use the convergence of c n ϕ n ( x ) in order to prove the convergence of c n ϕ n ( x ) thanks to relations in (2.31).

Proof. From (2.31)-(2.32) we get (2.34) as:

( n = 1 c n x ϕ n ( x ) = n = 1 c n α n ϕ n ( x ) [ 1 + A n ( x ) ] = n = 1 c n α n ϕ n ( x ) + n = 1 c n α n A n ϕ n ( x ) ; n = 1 c n | α n A n | ϕ n ( x ) A ( x ) n = 1 c n | α n | ϕ n ( x ) = o ( n = 1 c n α n ϕ n ( x ) ) . (2.36)

Case “ 0 < α + ”, which implies “ ϕ n ( x ) > 0 x n ”. First step:

lim ¯ x + f ( x ) x f ( x ) = lim ¯ x + n 1 c n ϕ n ( x ) n 1 c n x ϕ n ( x ) = lim ¯ x + n 1 c n ϕ n ( x ) n 1 c n α n ϕ n ( x ) [ 1 + A n ( x ) ] ( as 0 < 1 + A n ( x ) 1 + A ( x ) for x large enough ) lim ¯ x + n 1 c n ϕ n ( x ) [ 1 + A ( x ) ] n 1 c n α n ϕ n ( x ) ( as 0 < α n α ) lim ¯ x + 1 [ 1 + A ( x ) ] α = 1 α if α < + . (2.37)

Second step. Using a different device we get for each fixed k:

lim ¯ x + f ( x ) x f ( x ) lim ¯ x + n = 1 k 1 c n ϕ n ( x ) n = 1 c n x ϕ n ( x ) + lim ¯ x + n = k c n ϕ n ( x ) n = 1 c n x ϕ n ( x ) ( by the positivity of ϕ n ) lim ¯ x + n = 1 k 1 c n ϕ n ( x ) c k x ϕ k ( x ) + lim ¯ x + n = k c n ϕ n ( x ) n = k c n x ϕ n ( x ) . (2.38)

For the last two ratios we have:

n = 1 k 1 c n ϕ n ( x ) c k x ϕ k ( x ) = by (2.3) and (2.31) = o ( ϕ k ( x ) ) c k x ϕ k ( x ) = o ( 1 ) , x + ; (2.39)

R ( x ) : = n = k c n ϕ n ( x ) n = k c n x ϕ n ( x ) = n = k c n ϕ n ( x ) n = k c n α n ϕ n ( x ) [ 1 + o ( 1 ) ] 1 α k [ 1 + o ( 1 ) ] , x + , (2.40)

whence

lim ¯ x + R ( x ) 1 α k for each fixed k; (2.41)

and (2.38) yields:

lim ¯ x + f ( x ) x f ( x ) 1 α k for each fixed k. (2.42)

Taking the limit as k :

lim ¯ x + f ( x ) x f ( x ) ( 1 / α if 0 < α < + , 0 if α = + , (2.43)

whence:

lim x + f ( x ) x f ( x ) = 1 / α if 0 < α < + , by the inequality in (2.37); (2.44a)

lim x + f ( x ) x f ( x ) = 0 if α = + , as the ratio f ( x ) / x f ( x ) is ultimately >0; (2.44b)

and this last limit implies the limit “ + ” for the inverted ratio.

Case “ < α 0 ”, which, by (2.29) and (2.31), implies “ ϕ n ( x ) < 0 x n ”. Recall: α n < 0 and nondecreasing. First step:

x f ( x ) f ( x ) = ( 2.34 ) [ 1 + r ( x ) ] n = 1 c n α n ϕ n ( x ) n = 1 c n ϕ n ( x ) α [ 1 + r ( x ) ] , (2.45)

for x so large that “ 1 + r ( x ) > 0 ”; whence

lim _ x + x f ( x ) f ( x ) α . (2.46)

Second step. For each fixed k:

lim _ x + x f ( x ) f ( x ) = lim _ x + [ 1 + r ( x ) ] n = 1 c n α n ϕ n ( x ) n = 1 c n ϕ n ( x ) lim _ x + [ 1 + r ( x ) ] n = 1 k 1 c n α n ϕ n ( x ) n = 1 c n ϕ n ( x ) + lim _ x + [ 1 + r ( x ) ] n = k c n α n ϕ n ( x ) n = 1 c n ϕ n ( x ) . (2.47)

For the first ratio on the right we have “ | α n | | α n 1 | n ” and:

| n = 1 k 1 c n α n ϕ n ( x ) n = 1 c n ϕ n ( x ) | | α 1 | n = 1 k 1 c n ϕ n ( x ) n = 1 c n ϕ n ( x ) | α 1 | n = 1 k 1 c n ϕ n ( x ) c k ϕ k ( x ) = ( 2.3 ) o ( 1 ) , x + ; (2.48a)

whereas for the second ratio, as it stands without the absolute value, we have:

[ 1 + r ( x ) ] n = k c n α n ϕ n ( x ) n = 1 c n ϕ n ( x ) [ 1 + r ( x ) ] α n n = k c n ϕ n ( x ) n = 1 c n ϕ n ( x ) α k for x large enough. (2.48b)

From (2.47) we get:

lim _ x + x f ( x ) f ( x ) α k for each fixed k; (2.49)

and taking the limit as k :

lim _ x + x f ( x ) f ( x ) α , (2.50)

which, together with (2.46) yields “ lim x + x f ( x ) / f ( x ) = α ”.

Examples. Let { α n } n be a strictly increasing sequence of real numbers and c n > 0 n .

1) For ϕ n ( x ) : = x α n , x 1 , condition in (2.31) is trivially satisfied with A n ( x ) 0 .

First case. If lim n α n = α then:

n 1 c n x α n x α n 1 c n , x 1 ; n 1 c n | α n | x α n 1 ( sup | α n | ) x α 1 n 1 c n , x 1 ; (2.51)

and conditions in (2.30), (2.32) reduce to “ n 1 c n < + ” which implies (2.35). The identities “ x ϕ n ( k + 1 ) ( x ) ( α n k ) ϕ n ( k ) ( x ) k 0 ” and similar estimates for the formally differentiated series of higher order yield:

( n 1 c n < + , f ( x ) : = n 1 c n x α n ( f C [ 1 , + ) , f ( k ) R α k ( + ) k { 0 } . (2.52)

Consistently with the statement of Proposition 2.2 the circumstance “ α n k = 0 ” for some value of k and at most one value of n, say n = p , is treated by splitting the series

n 1 ( α n k ) ϕ n ( k ) ( x ) = ( n p + n > p ) ( α n k ) ϕ n ( k ) ( x ) .

Second case. If lim n α n = + then:

f ( x ) : = n 1 c n x α n < + x T > 0 ( f C [ T , + ) , f ( k ) R + ( + ) k . (2.53)

In fact, assuming α n > 0 n and with “ [ α n ] = integer part of α n ”, we have “ [ α n ] α n < [ α n ] + 1 ” and:

c n α n x α n = x c n α n x α n 1 x c n ( [ α n ] + 1 ) x [ α n ] ;

so that:

n 1 c n α n x α n x 2 n 1 c n [ α n ] x [ α n ] 1 + x n 1 c n x [ α n ] , (2.54)

where the second series on the right is a power series majorized by the convergent series n 1 c n x α n and the first series on the right is the formal derivative of a convergent power series, hence itself convergent uniformly on each compact subinterval of ] 0, + ) . Analogous situations for the higher-order derivatives and (2.53) follows.

2) For ϕ n ( x ) : = x α n ( log x ) β n , x T > 1 , we have

x ϕ n ( x ) = α n ϕ n ( x ) [ 1 + β n α n ( log x ) 1 ] , (2.55)

being α n 0 ultimately. Hence condition in (2.31) is satisfied if β n = O ( α n ) , n without any additional condition on β n .

The cases “ α n = constant ¯ ” deserve a separate brief discussion as they can be treated quite simply. The following results do not require any growth-order chain between the ϕ n ’s.

Proposition 2.3. (The case α n = constant ). Assume that:

L n C 1 [ T , + ) R 0 ( + ) ; c n > 0 n ; (2.56)

n = 1 c n L n ( x ) uniformly and absolutely convergent on each compact interval; (2.57)

x L n ( x ) = r n ( x ) L n ( x ) with | r n ( x ) | r ( x ) ( A x T , = o ( 1 ) , x + n . (2.58)

Then the function f ( x ) : = x α L n ( x ) , α satisfies “ f C 1 [ T , + ) R α ( + ) ”. Without either the restriction on the signs of c n or the conditions in (2.58) no definite conclusion on the type of variation of f can be drawn.

Proof. Notice that (2.56) implies L n ( x ) > 0 for x large enough and that we may always assume T > 0 . For α = 0 we have:

n 1 c n | L n ( x ) | x 1 r ( x ) n 1 c n | L n ( x ) | ( A T n 1 c n | L n ( x ) | , x T ; = o ( x 1 f ( x ) ) , x + ; (2.59)

whence it follows that

n = 1 c n L n ( x ) is uniformly and absolutely convergent on each compact interval of [ T , + ) .

Hence n = 1 c n L n ( x ) = f ( x ) x T and f ( x ) = o ( x 1 f ( x ) ) . For α 0 just apply the previous result to “ x α f ( x ) ” so obtaining “ ( x α f ( x ) ) = o ( x α 1 f ( x ) ) ”, whence “ f ( x ) α x 1 f ( x ) “ follows.

Example. Let { a n } n be any strictly decreasing sequence convergent to a 0 and define

ϕ n ( x ) : = ( log x ) a n , x e ; | x ϕ n ( x ) | = a n ( log x ) 1 ϕ n ( x ) a 1 ( log x ) 1 ϕ n ( x ) . (2.60)

If n 1 c n is any convergent series of constant positive terms then the function f ( x ) : = n 1 c n ϕ n ( x ) belongs to the class: f C 1 [ e , + ) R 0 ( + ) .

Counterexamples. Let L be any function such that:

L C 1 [ T , + ) R 0 ( + ) ; L ( + ) = + ; (2.61)

define ϕ n ( x ) : = ( L ( x ) ) n and let Φ ( x ) : = n 0 c n x n , c n 0 be an entire non-polynomial function. Here condition in (2.58) cannot be satisfied as x ϕ n ( x ) / ϕ ( x ) = n ( x L ( x ) / L ( x ) ) , except in case L ( x ) 0 , and the function f ( x ) : = n 0 c n ( L ( x ) ) n = Φ ( L ( x ) ) , x T , may have an arbitrarily large growth-order at + or may be slowly varying. To visualize, just consider the exponential power series and various elementary choices for L:

( n 0 ( c l 2 ( x ) ) n / n ! = ( log x ) c R 0 ( + ) , c 0 ; x [ e , + ) ; n 0 ( c ( log x ) δ ) n / n ! = exp ( c ( log x ) δ ) R 0 ( + ) , 0 < δ < 1 , c 0 ; x [ 1 , + ) ; n 0 ( c log x ) n / n ! = exp ( c log x ) x c R c ( + ) , c 0 ; x [ 1 , + ) ; n 0 ( ± ( log x ) δ ) n / n ! = exp ( ± ( log x ) δ ) R ± ( + ) , δ > 1 ; x [ 1 , + ) ; n 0 [ ± exp ( log x ) δ ] n / n ! = exp [ ± exp ( log x ) δ ] R ± ( + ) , 0 < δ < 1 ; x [ 1 , + ) . (2.62)

Proposition 2.4. (The case α n = ± ). Assumptions: (2.28) with α n = + n or α n = n , and c n > 0 n ; the uniform convergence of both series n = 1 c n ϕ n ( x ) , n = 1 c n ϕ n ( x ) on each compact interval. If α n = + and

M > 0 T M , independent of n, such that x ϕ n ( x ) > M ϕ n ( x ) x > T M n (2.63)

then f R + ( + ) . If α n = and

M < 0 T M , independent of n, such that x ϕ n ( x ) < M ϕ n ( x ) x > T M n (2.64)

then f R ( + ) .

Proof. Both proofs are trivial using the stated assumptions of “uniformity with respect to n” of the relations “ lim x + x ϕ n ( x ) / ϕ n ( x ) = ± ”. For α n = + :

M > 0 n = 1 c n x ϕ n ( x ) / n = 1 c n ϕ n ( x ) M x T M ,

and analogously for α n = .

For series, with positive coefficients, of functions having hypo-exponential, or exponential, or hyper-exponential variation at + corresponding results hold true suppressing the factor x on the left of the asymptotic relation in (2.31). All the proofs are exactly the same and, in particular, we have the result:

( ϕ n E α n ( + ) , < α n + , ϕ n ( x ) = ϕ n ( x ) [ α n + A n ( x ) ] , | A n ( x ) | A ( x ) = o ( 1 ) , x + , α n α , f E α ( + ) , (2.65)

provided that the remaining assumptions in Proposition 2.2 are satisfied.

Three more examples. The following examples are not included in the previous results but elucidate techniques based on the type of asymptotic variation.

1) Consider the sequence:

( ϕ n ( x ) : = exp [ ( log x ) 1 1 / n ] , x e , n 2 ; ϕ n R 0 ( + ) ; ϕ n 1 ( x ) ϕ n ( x ) x e ; ϕ n 1 ( x ) ϕ n ( x ) , x + , n 2 ; lim n ϕ n ( x ) = x uniformly on each compact subset of [ e , + ) ; ϕ n ( x ) = n 1 n x 1 ( log x ) 1 / n ϕ n ( x ) r n ( x ) ϕ n ( x ) > 0, x e . (2.66)

This is a sequence of slowly varying functions which is locally uniformly convergent to a regularly-varying function of index 1. The factor r n ( x ) satisfies:

0 < r n ( x ) x 1 , hence r n ( x ) = O ( x 1 ) uniformly with respect to n; (2.67)

but, though “ r n ( x ) = o ( x 1 ) , x + ” for each fixed n, this last relation is not uniform with respect to n. Suppose now that for a sequence of positive numbers c n the series n 2 c n ϕ n ( x ) is locally uniformly convergent; then

0 < n 2 c n ϕ n ( x ) n 2 c n ϕ n ( x ) ;

hence the series of the derivatives is locally uniformly convergent as well and, by (2.65), f E o ( + ) . But direct calculations yield more precise information:

f ( x ) ( = x 1 n 2 n 1 n c n ( log x ) 1 / n ϕ n ( x ) x 1 n 2 c n ϕ n ( x ) = x 1 f ( x ) , x 1 ( log x ) 1 n 2 n 1 n c n ϕ n ( x ) 1 2 x 1 ( log x ) 1 n 2 c n ϕ n ( x ) = 1 2 x 1 ( log x ) 1 f ( x ) , (2.68)

valid for x e ; whence, integrating the ratio f / f :

log ( f ( x ) / f ( e ) ) ( log ( x / e ) 1 2 l 2 ( x ) , x e , (2.69a)

f ( x ) ( ( f ( e ) / e ) x = 1 e ( n 2 c n ϕ n ( e ) ) x = ( n 2 c n ) x e ( n 2 c n ) ( log x ) 1 / 2 , x e . (2.69b)

The relations in the first line of (2.68) imply the two limits

lim _ x + x f ( x ) / f ( x ) 0, lim ¯ x + x f ( x ) / f ( x ) 1 ;

hence, if f happens to be regularly varying of some index α , then 0 α 1 .

2) Consider the sequence:

( ϕ n ( x ) : = exp [ ( log x ) n ] , x e , n 2 ; ϕ n R + ( + ) ; ϕ n 1 ( x ) ϕ n ( x ) x e ; ϕ n 1 ( x ) ϕ n ( x ) , x + , n 2 ; ϕ n ( x ) = n x 1 ( log x ) n 1 ϕ n ( x ) , x e . (2.70)

If the two conditions in (2.30) hold true then:

f ( x ) = x 1 n 2 n c n ( log x ) n 1 ϕ n ( x ) x 1 ( log x ) n 2 c n ϕ n ( x ) = x 1 ( log x ) f ( x ) ; ( c n > 0 ) ;

whence: x f ( x ) / f ( x ) + i.e. f R + ( + ) . The inequality “ f ( x ) / f ( x ) x 1 log x , x e ” yields the global estimates:

( log ( f ( x ) / f ( e ) ) e x t 1 log t d t = 1 2 ( log x ) 2 1 2 , x e ; f ( x ) f ( e ) e 1 / 2 exp [ ( log x ) 2 / 2 ] = e 1 / 2 exp [ ( log x ) 2 / 2 ] , x e ; (2.71)

and obviously:

f ( x ) c k ϕ k ( x ) x , k and f ( x ) = + ( exp ( log x ) k ) , x + , for each fixed k 2. (2.72)

3) For a bounded sequence of real numbers of arbitrary signs, | c n | M , consider the following two series:

f ( x ) : = n 1 c n 1 + ( n x ) α , f ( x ) : = α x α 1 n 1 c n n α ( 1 + ( n x ) α ) 2 , α > 1 , (2.73)

which are absolutely and uniformly convergent on the whole interval [ 1, + ) because:

( n 1 c n 1 + ( n x ) α = x α n 1 c n n α ( 1 + ( n x ) α ) x α n 1 c n n α ; n 1 c n n α ( 1 + ( n x ) α ) 2 = x 2 α n 1 c n n 2 α ( 1 + ( n x ) α ) 2 x 2 α n 1 c n n α . (2.74)

We also have the estimates:

( 1 2 x α n 1 c n n α f ( x ) x α n 1 c n n α , x 1 ; α 4 x α 1 n 1 c n n 2 α | f ( x ) | α x α 1 n 1 c n n 2 α α x α 1 n 1 c n n α , x 1 ; (2.75)

hence f is positive, strictly decreasing and:

2 α x f ( x ) / f ( x ) α / 4 , (2.76)

which implies:

λ 2 α f ( λ x ) f ( x ) λ α / 4 fixed λ > 1, (2.77)

a weaker concept of regular variation thoroughly studied in ( [4]; Ch. 2).

3. Type of Asymptotic Variation of a Wronskian

In two papers [5] [6] the author described a number of techniques to obtain the asymptotic behaviors of Wronskians whose entries are regularly- or rapidly-varying functions of higher order; here we point out some cases wherein one can specify the type of asymptotic variation of the involved Wronskians under the natural additional assumption on the nth derivatives. Proofs are based on the evaluation of a type of Vandermonde determinants with a gap in the exponents, determinants which are still a part of the classical theory of determinants.

Lemma 3.1. (I) If ( c 1 , , c n ) is an ordered n-tuple of complex numbers with n 2 , its Vandermondian with a one-unit gap in the highest exponent is defined as the number:

V ˜ ( c 1 , c 2 ) : = | 1 1 c 1 2 c 2 2 | = ( c 2 2 c 1 2 ) = V ( c 1 , c 2 ) ( c 1 + c 2 ) ; (3.1)

V ˜ ( c 1 , , c n ) : = | 1 1 1 c 1 c 2 c n c 1 2 c 2 2 c n 2 c 1 n 2 c 2 n 2 c n n 2 c 1 n c 2 n c n n | = V ( c 1 , , c n ) ( c 1 + + c n ) , n 3 , (3.2)

where V ( c 1 , , c n ) denotes the standard Vandermondian.

(II) The following formula holds true:

| 1 1 c 1 c n ( c 1 ) 2 _ ( c n ) 2 _ ( c 1 ) n 2 _ ( c n ) n 2 _ ( c 1 ) n _ ( c n ) n _ | = V ( c 1 , , c n ) ( c 1 + + c n n ( n 1 ) / 2 ) , n 2. (3.3)

Proof. Part I is mentioned as an exercise by Mirsky ( [7]; Problem 30 on Chapter I, p. 38): it can be proved by polynomial algebra adapting the argument in ( [7]; p. 18). Part II is checked at once for n = 2 and can be proved for n 3 by repeating, as a first step, the elementary procedure used in reducing the determinant det [ 1 , c i , ( c i ) 2 _ , , ( c i ) n _ ] i = 1 n to the Vandermondian: this is sketched in ( [5]; Lemma 4 part 5, p. 10) and then used twice, in ( [5]; proof of Th. 9, p. 19) and ( [6]; proof of Th. 3, p.19). In the present case the procedure, when iterated until the exponent n 2 , yields:

| 1 c i ( c i ) 2 _ ( c i ) n 2 _ ( c i ) n _ | i = 1 , , n = | 1 c i c i 2 c i n 2 ( c i ) n _ | i = 1 , , n , (3.4)

where, as easily checked, the polynomial in c i , ( c i ) n _ , has an expression of the form

( c i ) n _ c i n n ( n 1 ) 2 c i n 1 + a n 2 c i n 2 + + a 1 c 1 ,

with coefficients a k independent of the index i so that, by subtracting the appropriate linear combination of the preceding rows, we get that the determinant on the right in (3.4) equals

| 1 c i c i 2 c i n 2 c i n n ( n 1 ) 2 c i n 1 | i = 1 , , n = | 1 c i c i 2 c i n 2 c i n | i = 1 , , n n ( n 1 ) 2 | 1 c i c i 2 c i n 2 c i n 1 | i = 1 , , n = V ˜ ( c 1 , , c n ) n ( n 1 ) 2 V ( c 1 , , c n ) , (3.5)

and (3.3) follows.

The following is a complement to Theorem 9 in ( [5]; p. 18).

Theorem 3.2. (Wronskians of smoothly-varying functions). Let ϕ i C n [ T , + ) , ϕ i ( x ) 0 for x large enough, 1 i n , n 2 , and

ϕ i ( k ) ( x ) / ϕ i ( x ) = x k [ ( a i ) k _ + o ( 1 ) ] , x + , i , k { 0 , 1 , , n } ; a i a j i j ; (3.6)

which means that each ϕ i is smoothly varying of order n and index a i according to Definition 3.2 in ( [1]; p. 803). Then:

( W ( x ) : = W ( ϕ 1 ( x ) , , ϕ n ( x ) ) ~ V ( a 1 , , a n ) ( i = 1 n ϕ i ( x ) ) x n ( n 1 ) / 2 , W ( x ) / W ( x ) = x 1 [ a 1 + + a n n ( n 1 ) / 2 + o ( 1 ) ] , x + , (3.7)

i.e. W is regularly varying of order n and index “ a 1 + + a n n ( n 1 ) / 2 ” in the strong sense of Definition 2.1 in ( [1]; p. 781). In particular W is slowly varying whenever a 1 + + a n = n ( n 1 ) / 2 such as the elementary Wronskian

W ( x a 1 , , x a n ) = V ( a 1 , , a n ) = constant ”: ( [5]; formula (68), p. 10).

Proof. The first relation in (3.7) is proved in ( [5]; Theorem 9, p. 18) whereas for the behavior of W we have:

W ( x ) | ϕ i ( x ) ϕ i ( x ) ϕ i ( n 2 ) ( x ) ϕ i ( n ) ( x ) | i = 1 , , n = ( i = 1 n ϕ i ( x ) ) | 1 x 1 [ a i + o ( 1 ) ] x 2 [ ( a i ) 2 _ + o ( 1 ) ] x n 2 [ ( a i ) n 2 _ + o ( 1 ) ] x n [ ( a i ) n _ + o ( 1 ) ] | i = 1 , , n = ( i = 1 n ϕ i ( x ) ) x n ( n 1 ) 2 1 | 1 [ a i + o ( 1 ) ] [ ( a i ) 2 _ + o ( 1 ) ] [ ( a i ) n 2 _ + o ( 1 ) ] [ ( a i ) n _ + o ( 1 ) ] | i = 1 , , n . (3.8)

Applying the procedure in (3.4)-(3.5) it is seen that the last determinant equals “ V ˜ ( c 1 , , c n ) + o ( 1 ) ”, and by Lemma 3.1 we get:

W ( x ) = ( i = 1 n ϕ i ( x ) ) x n ( n 1 ) 2 1 [ V ( a 1 , , a n ) ( a 1 + + a n n ( n 1 ) / 2 ) + o ( 1 ) ] , x + ; (3.9)

whence the second relation in (3.7) follows.

To obtain a result in the case of rapid variation we need a correct statement of the analogue of Theorem 6 in ( [5]; pp. 11-12) for the determinant V ˜ . Rereading the proof of this theorem we get the following claims.

Lemma 3.3. Let g 1 , , g n and f 1 , , f n be functions defined on a deleted neighborhood of x 0 ¯ and such that:

g i ( x ) = f i ( x ) + o ( f i ( x ) ) , x x 0 , 1 i n , n 2. (3.10)

The following relations hold true. (I) The general estimate

V ˜ ( g 1 ( x ) , , g n ( x ) ) V ˜ ( f 1 ( x ) + o ( f 1 ( x ) ) , , f n ( x ) + o ( f n ( x ) ) ) = V ˜ ( f 1 ( x ) , , f n ( x ) ) + o ( ( p 1 , , p n ) P i = 1 n | f i ( x ) | p i ) , x x 0 , (3.11)

where P denotes the set of all permutations of the n-tuple ( 0,1, , n 2, n ) . Formula (3.11) must be read with the agreement that “ | f i ( x ) | 0 1 ” regardless of the possible zeros of f i ( x ) . For n = 2 it reduces to:

V ˜ ( f 1 ( x ) + o ( f 1 ( x ) ) , f 2 ( x ) + o ( f 2 ( x ) ) ) = V ˜ ( f 1 ( x ) , f 2 ( x ) ) + o ( | f 1 ( x ) | 2 + | f 2 ( x ) | 2 ) , x x 0 . (3.12)

(II) If we assume the following relationships between the f i ’s:

f 1 ( x ) f 2 ( x ) f n ( x ) , x x 0 , i , (3.13a)

[where “ f g ” means “ g = O ( f ) ”], in practice:

either f i ( x ) f i + 1 ( x ) or f i ( x ) f i + 1 ( x ) , x x 0 , i , (3.13b)

then relation (3.11) takes the simpler form:

V ˜ ( f 1 ( x ) + o ( f 1 ( x ) ) , , f n ( x ) + o ( f n ( x ) ) ) = V ˜ ( f 1 ( x ) , , f n ( x ) ) + o ( f 1 ( x ) i = 1 n 1 | f i ( x ) | n i ) , x x 0 , (3.14)

noticing, in the right-hand side, the lack of f n which is one of the functions f i with the lowest growth-order. The two most meaningful cases are highlighted in the following statements.

(III) If ( f 1 , , f n ) is an asymptotic scale, i.e. f 1 ( x ) f n ( x ) , x x 0 , then (3.14) becomes

V ˜ ( f 1 ( x ) + o ( f 1 ( x ) ) , , f n ( x ) + o ( f n ( x ) ) ) = ( 1 ) n ( n 1 ) / 2 ( f 1 ( x ) i = 1 n 1 ( f i ( x ) ) n i ) + o ( f 1 ( x ) i = 1 n 1 | f i ( x ) | n i ) , x x 0 . (3.15)

(IV) If all the f i ’s have the same growth-order in the sense that

f i ( x ) = c i f ( x ) + o ( f ( x ) ) , x x 0 , 1 i n , (3.16)

for a fixed f and arbitrary pairwise-distinct constants c i , then:

V ˜ ( c 1 f ( x ) + o ( f ( x ) ) , , c n f ( x ) + o ( f ( x ) ) ) = V ( c 1 , , c n ) ( c 1 + + c n ) ( f ( x ) ) n ( n 1 ) 2 + 1 + o ( | f ( x ) | n ( n 1 ) 2 + 1 ) , x x 0 . (3.17)

In the special case f ( x ) 1 we get:

V ˜ ( c 1 + o ( 1 ) , , c n + o ( 1 ) ) | 1 1 c 1 + o ( 1 ) c n + o ( 1 ) c 1 2 + o ( 1 ) c n 2 + o ( 1 ) c 1 n 2 + o ( 1 ) c n n 2 + o ( 1 ) c 1 n 1 + o ( 1 ) c n n 1 + o ( 1 ) | = V ( c 1 , , c n ) ( c 1 + + c n ) + o ( 1 ) , x x 0 , (3.18)

a relation already used in the proof of Theorem 3.2.

Hints for the proof. When rereading Theorem 6 in ( [5]; pp. 11-14) the reader will notice that the agreement “ | f i ( x ) | 0 1 ” is not explicitly stated in the statement but it is clearly specified in the proof ( [5]; p. 13, line 8 from below). At this point of the proof in [5] the estimate in (3.11) is proved; to proceed, the reader will replace the quantity in formula (95) in ( [5]; p. 13) with:

| f 1 ( x ) | n | f 2 ( x ) | n 2 | f n 1 ( x ) | 1 | f n ( x ) | 0 , (3.19)

which stands for one of the terms with maximal growth-order appearing in the sum inside the “o”-term in (3.11), an assertion justified by noticing that an interchange of exponents in two factors in (3.19), leaving unchanged the other factors, does not increase the growth-order:

( for the product | f i | n i | f j | n j , replaced by | f i | n j | f j | n i , 2 i < j , we have : | f i | n j | f j n i | = | f i | n i | f j n j | | f i | i j | f j j i | = | f i | n i | f j n j | [ | f i | i j O ( | f i j i | ) ] = ( | f i | n i | f j n j | ) O ( 1 ) , ( i < j ) ; (3.20)

( for the product | f 1 | n | f j | n j , replaced by | f 1 | n j | f j | n , j > 1 , we have : | f 1 | n j | f j | n = | f 1 | n | f j n j | | f 1 | j | f j j | = | f 1 | n | f j | n j O ( 1 ) , ( j > 1 ) . (3.21)

This argument is valid under condition (3.13a) and is independent of the possible zeros of the f i ’s unlike the original device used in ( [5]; formula (96), p. 13) as pointed out in ( [6]; §2, Comments on Theorem 6, part (C), p. 39). The subsequent reasonings in the proof of Theorem 6 in [5] remain unchanged with the only obvious change of the expression of V ( f 1 ( x ) , , f n ( x ) ) with that of V ˜ ( f 1 ( x ) , , f n ( x ) ) in formulas (97)-(98) in ( [5]; p. 14).

The following is a complement to Theorem 10 in ( [5]; pp. 19-20).

Theorem 3.4. (Wronskians of rapidly-varying functions). Consider n functions ϕ i C n [ T , + ) , i = 1 , , n with both ϕ i ( x ) and ϕ i ( x ) 0 for x large enough, 1 i n , n 2 ; and let them satisfy the following relations:

ϕ i ( k ) ( x ) / ϕ i ( x ) ( ϕ i ( x ) / ϕ i ( x ) ) k , x + ; 1 i , k n , (3.22)

which, in particular, are satisfied by functions which are rapidly varying at + of order n 1 in the strong restricted sense of ( [5]; Def. 4.1, p. 807, and Prop. 4.1, p. 808).

(I) If

ϕ 1 ( x ) / ϕ 1 ( x ) ϕ n ( x ) / ϕ n ( x ) , x + , (3.23)

then, as x + :

( W ( x ) : = W ( ϕ 1 ( x ) , , ϕ n ( x ) ) ( 1 ) n ( n 1 ) / 2 ( i = 1 n ϕ i ( x ) ) ( i = 1 n 1 ( ϕ i ( x ) ϕ i ( x ) ) n i ) , W ( x ) / W ( x ) ϕ 1 ( x ) / ϕ 1 ( x ) , (3.24)

which means that W has the same type of asymptotic variation as ϕ 1 according to the concept first formulated by Hardy [8] and, in particular, W has the same type of exponential variation as ϕ 1 according to ( [2]; Def. 8.1, p. 832).

(II) If

ϕ 1 ( x ) / ϕ 1 ( x ) = c i ϕ ( x ) + o ( ϕ ( x ) ) , x + , i = 1, , n , (3.25)

for some fixed function ϕ and pairwise-distinct constants c i then, as x + :

( W ( x ) : = W ( ϕ 1 ( x ) , , ϕ n ( x ) ) V ( c 1 , , c n ) ( i = 1 n ϕ i ( x ) ) ( ϕ ( x ) ) n ( n 1 ) / 2 , W ( x ) / W ( x ) = ( c 1 + + c n ) ϕ ( x ) + o ( ϕ ( x ) ) . (3.26)

If, moreover, all the c i ’s and their sum ( c 1 + + c n ) are non-zero then:

1) if “ lim x + ϕ ( x ) = either 0 or ± ”, W has the same type of hypo-exponential or hyper-exponential variation as all the ϕ i ’s;

2) if “ lim x + ϕ ( x ) \ { 0 } ”, relations in (3.25) may be written (possibly changing the constants) as:

ϕ i ( x ) / ϕ i ( x ) = c i + o ( 1 ) , x + , i = 1, , n , i . e . ϕ i E c i ( + ) , (3.27)

and W E c 1 + + c n ( + ) . If the c i ’s are pairwise-distinct non-zero numbers and c 1 + + c n = 0 then W is hypoexponentially varying such as the Wronskian “ W ( e a 1 x , , e a n x ) = V ( a 1 , , a n ) = constant ”: ( [5]; formula (68), p. 10, case g ( x ) e x ).

Proof. The asymptotic relations for W are given in ( [5]; pp. 19-20); those for W and n 3 —the case n = 2 being trivial—follow from Lemma 3.3 and the simple calculations in ( [5]; proof of Th. 10, p. 21) in the two cases. In fact, under conditions in (3.23):

W ( x ) | ϕ i ( x ) ϕ i ( x ) ϕ i ( n 2 ) ( x ) ϕ i ( n ) ( x ) | i = 1, , n = ( i = 1 n ϕ i ( x ) ) | 1 ϕ i ( x ) / ϕ i ( x ) ( ϕ i ( x ) / ϕ i ( x ) ) 2 [ 1 + o ( 1 ) ] ( ϕ i ( x ) / ϕ i ( x ) ) n 2 [ 1 + o ( 1 ) ] ( ϕ i ( x ) / ϕ i ( x ) ) n [ 1 + o ( 1 ) ] | i = 1, , n ( 3.15 ) ( 1 ) n ( n 1 ) / 2 ( i = 1 n ϕ i ( x ) ) ( i = 1 n 1 ( ϕ i ( x ) ϕ i ( x ) ) n i ) ϕ 1 ( x ) ϕ 1 ( x ) , x + ; (3.28)

and under conditions in (3.25):

W ( x ) V ( c 1 , , c n ) ( c 1 + + c n ) ( ϕ ( x ) ) n ( n 1 ) 2 + 1 , x + . (3.29)

4. Asymptotic Differential Equations versus Asymptotic Functional Equations

The theory developed in [1] [2] concerns functions differentiable a certain number of times and it starts from various types of asymptotic relations; for instance, the relation

x f ( x ) / f ( x ) = α + o ( 1 ) , x + , (4.1a)

defines the basic concept of regular variation of index α in the strong sense, a concept (but not the locution) dating back to Hardy [8] . The above relation may be termed an “asymptotic differential equation”. But, as mentioned in ( [1]; p. 782), a larger class of functions with substantially the same fundamental properties was first introduced by Karamata, and then extensively studied by other mathematicians, starting from the “asymptotic functional equation”

f ( λ x ) λ α f ( x ) , x + , for each fixed λ > 0 , with f measurable and positive. (4.1b)

It is known that for a function f with a monotonic derivative conditions (4.1a) and (4.1b) are equivalent and we show in this section that the same is true for the pair of equations (differential and functional) pertinent to each of the other types of variation: rapid, hypo-exponential, exponential or hyper-exponential. The first result of this type has been proved by Lamperti ( [9]; pp. 382-383) for everywhere-differentiable functions using the Lagrange mean-value formula: f ( x ) f ( y ) = f ( c ) ( x y ) . The integral version of the mean-value formula provides proofs for absolutely continuous functions when suitably reading the asymptotic relations for the derivative. For the sake of completeness, we make explicit some elementary remarks about condition “ f monotonic” in three possible interpretations.

­ If “ f exists everywhere on an interval I and is monotonic thereon” then an elementary argument shows that f is absolutely continuous on each compact subinterval of I ( [10]; Problem A, p. 13), and obviously is either concave or convex on I.

­ If “ f exists on I save possibly a countable set N and is monotonic on I\N” then the above conclusion about absolute continuity follows as a corollary of a non-trivial result in the Lebesgue theory ( [11]; p. 299); and the concavity-convexity character follows from the classical characterization ( [10]; Ch. 12, Th. A, pp. 9-10) or ( [11]; Ch. V (18.43), p. 300), via an integral representation:

f ( x ) = f ( c ) + c x ϕ ( t ) d t ; x , c I , with ϕ monotonic on I , (4.2)

and where the role of ϕ may be played indifferently by the left or the right derivative of f , both existing everywhere and coinciding except possibly on a countable set: ϕ non-decreasing for convexity and non-increasing for concavity.

­ If “ f exists on I save possibly a set N ˜ of Lebesgue measure zero and is monotonic on I \ N ˜ and if the absolute continuity of f is explicitly assumed” then for any three numbers in I, x < z < y , and, say, f non-decreasing we have the inequalities:

f ( z ) f ( x ) z x = 1 z x x z f sup [ x , z ] \ N ˜ f inf [ x , y ] \ N ˜ f 1 y z z y f = f ( y ) f ( z ) y z , (4.3)

whence, by standard calculations, we get the inequality:

f ( z ) f ( x ) z x f ( y ) f ( x ) y x , x < z < y . (4.4)

In fact:

f ( z ) f ( x ) z x f ( y ) f ( x ) y x 0 ( y x ) ( f ( z ) f ( x ) ) ( z x ) ( f ( y ) f ( x ) ) 0 writing y x = y z + z x ( y z ) ( f ( z ) f ( x ) ) + ( z x ) ( f ( z ) f ( x ) f ( y ) + f ( x ) ) 0 ( y z ) ( f ( z ) f ( x ) ) + ( z x ) ( f ( z ) f ( y ) ) 0,

which is true by (4.3). This shows that the difference quotient with a fixed endpoint is non-decreasing, hence the left and right derivatives exists as finite numbers at each interior point of I and are non-decreasing. Analogously for f non-increasing. These remarks show that:

In the setting of absolute continuity, condition “ f monotonic” even in its weakest meaning unambiguously refers to a function whose left and right derivatives exist as finite numbers at each interior point, coincide except possibly on a countable set and are monotonic with the appropriate type of monotonicity.

Moreover, an asymptotic relation involving f such as, for instance (4.1a) may be read in any of the following three ways:

( x f + ( x ) / f ( x ) = α + o ( 1 ) , x + , x [ T , + ) ; x f ( x ) / f ( x ) = α + o ( 1 ) , x + , x [ T , + ) ; x f ( x ) / f ( x ) = α + o ( 1 ) , x + , x [ T , + ) \ { a suitable countable set } ; (4.5)

whereas, strictly speaking, the shortened notation in (4.1a) refers to the case of an everywhere-differentiable function. The following two theorems show that the differential and the functional approaches to the theory of asymptotic variation coincide for functions which are ultimately either concave or convex which is certainly the case of functions whose first derivatives are regularly varying of index 0 or rapidly varying. In the proofs use is made of the integral representation in (4.2).

Theorem 4.1. (Regular and rapid variation). For a positive function f either concave or convex on an interval [ T , + ) the following equivalences hold true:

{ f ( λ x ) λ α f ( x ) , x + , for each fixed λ > 0 and a constant α } x f ( x ) / f ( x ) = α + o ( 1 ) , x + ; (4.6)

f ( λ x ) = ( o ( f ( x ) ) if 0 < λ < 1 , + ( f ( x ) ) if λ > 1 , x f ( x ) / f ( x ) = + ( 1 ) , x + ; (4.7)

f ( λ x ) = ( + ( f ( x ) ) if 0 < λ < 1 , o ( f ( x ) ) if λ > 1 , x f ( x ) / f ( x ) = ( 1 ) , x + . (4.8)

Theorem 4.2. (Types of exponential variation). For a positive function f either concave or convex on an interval [ T , + ) the following equivalences hold true:

{ f ( x + λ ) f ( x ) , x + , for each fixed λ } f ( x ) = o ( f ( x ) ) , x + ; (4.9)

{ f ( x + λ ) e c λ f ( x ) , x + , for each fixed λ and c 0 } f ( x ) = f ( x ) [ c + o ( 1 ) ] , x + ; (4.10)

f ( x + λ ) = ( o ( f ( x ) ) if λ < 0 , + ( f ( x ) ) if λ > 0 , f ( x ) / f ( x ) = + ( 1 ) , x + ; (4.11)

f ( x + λ ) = ( + ( f ( x ) ) if λ < 0 , o ( f ( x ) ) if λ > 0 , f ( x ) / f ( x ) = ( 1 ) , x + . (4.12)

Proof of Theorem 4.1. All the inferences from the right to the left in both theorems elementarily follow from the integral representations and are to be found in ( [1]; §5) and in ( [2]; §8). Here we have to prove the converses. All the proofs are based on estimating the integral x y f f ( y ) f ( x ) in terms of f where f indifferently stands for the right or left derivative of f which both exist everywhere. Proof of (4.6); for f non-decreasing we have:

f ( x ) 1 ( λ 1 ) x x λ x f f ( λ x ) f ( x ) ( λ 1 ) x f ( λ x ) , λ > 1 , (4.13)

whence:

x f ( x ) f ( x ) 1 λ 1 ( f ( λ x ) f ( x ) 1 ) x f ( λ x ) f ( x ) λ x f ( λ x ) f ( λ x ) λ 1 f ( λ x ) f ( x ) , λ > 1 ; (4.14)

and letting x + :

lim ¯ x + x f ( x ) f ( x ) λ α 1 λ 1 λ α 1 lim _ x + x f ( x ) f ( x ) , λ > 1 ; (4.15)

and we get our claim letting λ 1 + . For f non-increasing use the same argument reversing the inequalities in (4.13). For the proof of (4.7) we have:

f non-decreasing f ( λ x ) f ( λ x ) f ( x ) ( λ 1 ) x , λ > 1, λ x f ( λ x ) f ( λ x ) λ λ 1 ( 1 f ( x ) f ( λ x ) ) , λ > 1, as x + lim _ y + y f ( y ) f ( y ) λ λ 1 , λ > 1 ; (4.16)

and then let λ 1 + . If f would be non-increasing (see however Remark 2 about growth-estimates after the proof of Theorem 4.2) then we might more simply write:

f ( x / 2 ) f ( x ) f ( x / 2 ) x / 2 whence ( x / 2 ) f ( x / 2 ) f ( x / 2 ) ( f ( x ) f ( x / 2 ) 1 ) = + ( 1 ) . (4.17)

The result in (4.8) is brought back to (4.7) putting g ( x ) : = 1 / f ( x ) .

Proof of Theorem 4.2. Notice that both the functional equations and the differential equations listed in Theorem 4.2 may be brought back to the corresponding ones in Theorem 4.1 by a change of variable; in fact, putting g ( x ) : = f ( e x ) , where f ( x ) 0 for x large enough, we have:

( f ( x ) = g ( log x ) ; f ( λ x ) = g ( log x + log λ ) , λ > 0 ; lim x + [ f ( λ x ) / f ( x ) ] = lim x + [ g ( log x + log λ ) / g ( log x ) ] = lim y + [ g ( y + log λ ) / g ( y ) ] ϕ ( λ ) , λ > 0 ; hence : lim y + [ g ( y + λ ) / g ( y ) ] = ϕ ( e λ ) , λ ; (4.18a)

and the following are easily checked:

lim x + f ( λ x ) f ( x ) = ( 1 for λ > 0 , λ α for λ > 0 , 0 for 0 < λ < 1 , + for λ > 1 , lim x + g ( x + λ ) g ( x ) = ( 1 for λ , e α λ for λ , 0 for λ < 0 , + for λ > 0 ; (4.18b)

lim x + x f ( x ) f ( x ) = l ¯ lim x + g ( x ) g ( x ) = l . (4.18c)

But a correspondence between the monotonicity of f and g is not granted; hence it is better to give direct proofs of Theorem 4.2 following the same patterns as above. We write them down only for “ f non-decreasing”. For the proof of (4.9) notice that the claim does not depend on the sign of f ; hence we may suppose “ f 0 ” changing, if necessary, the sign of f . For “ f non-decreasing” we have:

0 f ( x ) f ( x + 1 ) f ( x ) = f ( x ) + o ( f ( x ) ) f ( x ) = o ( f ( x ) ) . (4.19)

For the proof of (4.10) and “ f non-decreasing” we have:

f ( x ) f ( x + λ ) f ( x ) λ f ( x + λ ) , λ > 0 ; (4.20)

whence:

f ( x ) f ( x ) 1 λ ( f ( x + λ ) f ( x ) 1 ) f ( x + λ ) f ( x ) f ( x + λ ) f ( x + λ ) f ( x + λ ) f ( x ) , λ > 0 ; (4.21)

and as x + :

lim ¯ x + f ( x ) / f ( x ) 1 λ ( e c λ 1 ) e c λ lim _ x + f ( x + λ ) / f ( x + λ ) = e c λ lim _ x + f ( x ) / f ( x ) , λ > 0 ; (4.22)

and for λ 0 + :

lim ¯ x + f ( x ) / f ( x ) c lim _ x + f ( x ) / f ( x ) . (4.23)

And for the proof of (4.11) we write:

f ( x + λ ) f ( x + λ ) f ( x ) λ , hence f ( x + λ ) f ( x + λ ) 1 λ ( 1 f ( x ) f ( x + λ ) ) , λ > 0, (4.24)

and the assertion follows by taking first the “lim inf” as x + and then the limit as λ 0 + .

Remarks about growth-estimates. 1) One of the basic properties of the functions in the studied classes is the estimates, though rough, of their growth-orders. For the subclasses defined via asymptotic differential equations the estimates are elementarily inferred from the pertinent integral representations whereas for the general classes it is true that they are inferred from suitable integral representations as well, but such representations are not elementary facts but consequences of the nontrivial core of the theory, namely the “uniform convergence theorems”. For slow (hence for regular) variation the estimates are explicitly mentioned in ( [4]; Prop.1.3.6, p.16):

f satisfying the functional equation in (4.1b)” x α ϵ f ( x ) x α + ϵ , x + , ϵ > 0. (4.25)

For rapid variation a “representation theorem” ( [4]; Prop. 2.4.4, p. 85, and Th. 2.4.5, p. 86), can be obtained only for a certain subclass; in particular, if f is monotonic on an interval [ T , + ) the two functional equations in (4.7) imply that f is non-decreasing and admits of the integral representation:

f ( x ) = exp { z ( x ) + η ( x ) + T x t 1 ξ ( t ) d t } , x T , (4.26)

where the measurable functions z , η , ξ are such that: z is non-decreasing, η ( x ) 0 and ξ ( x ) + as x + (This is the case of Theorem 4.1 wherein the monotonicity of f implies the ultimate monotonicity of f ). From (4.26) the following estimate is trivially inferred:

x α f ( x ) A exp ( T x t 1 [ ξ ( t ) α ] d t ) , ( A = constant > 0 ) , (4.27a)

whence

f ( x ) x α , x + , α > 0. (4.27b)

By putting g ( x ) : = f ( e x ) one gets the corresponding growth-estimates for exponential variations listed in ( [2]; §8).

2) In the case of (4.7) the contingency “ f non-increasing” is excluded otherwise we would have:

lim x + f ( x ) / x = lim x + f ( x ) = ( a , (4.28)

contradicting either the growth-estimate in (4.27b) or the positivity of f .

3) In passing we point out that the above-mentioned “uniform convergence theorems” imply that a function satisfying one of the asymptotic functional equation mentioned in Theorems 4.1-4.2 satisfies a quite stronger functional equation. A list appears in ( [1]; §5) inferred from the simple integral representations valid when the corresponding asymptotic differential equations hold true: the pertinent calculations are elementary, though not trivial, implicitly using the uniform convergence with respect to the parameter. In the more general context we have:

( f ( λ x ) λ α f ( x ) , for each λ > 0, f measurable , without any monotonicity restriction , f ( λ x + o ( x ) ) λ α f ( x ) , x + , α ; (4.29)

a result inferred from the factorization f ( x ) = x α l ( x ) , with l slowly varying, and from ( [4]; Th. 1.2.1, p. 6). And we also have:

( f ( λ x ) = ( o ( f ( x ) ) if 0 < λ < 1 , + ( f ( x ) ) if λ > 1 , f measurable , without any monotonicity restriction ,

f ( λ x + o ( x ) ) = ( o ( f ( x ) ) if 0 < λ < 1 , + ( f ( x ) ) if λ > 1 , , x + ; (4.30)

inferred from ( [4]; Cor. 2.4.2, p.85).

The reader may write down the corresponding versions of Theorems 4.1-4.2 for the functional equations listed in ( [1]; Prop. 5.2, pp. 814-815) under the monotonicity of f .

5. Concepts Related to Logarithmic Variation

The concept of regular variation of order α 0 generalizes the asymptotic behavior of a power whereas the three concepts of exponential variation generalize the asymptotic behaviors of the exponential of a power. These generalizations are quite natural whereas the concept of slow variation is not the appropriate generalization of the behavior of the logarithm: it shares some asymptotic properties of the logarithm but it encompasses functions with orders of growth either greater than the order of each positive power of the logarithm or less than the order of each negative power of the logarithm such as the functions: exp [ ± ( log x ) ] δ , 0 < δ < 1 . Looking at the asymptotic functional equation of a slowly-varying function l ,

l ( λ x ) l ( x ) , x + , (5.1a)

we see no link with the parameter λ , whereas the arithmetic functional equation characterizing the logarithm,

l ( λ x ) = l ( x ) + l ( λ ) λ , x > 0 , (5.1b)

if interpreted asympotically, gives an expression for the remainder. One of the possible asymptotic counterparts of (5.1b), as x + , is the asymptotic functional equation:

f ( λ x ) = f ( x ) + ϕ ( λ ) + o ( 1 ) , x + , fixed λ > 0. (5.2)

It is known, ( [4]; Lemma 3.2.1-case g ( x ) 1 , p. 140) that if (5.2) is satisfied by a measurable function f defined on a neighborhood of + then “ ϕ ( λ ) = c log λ λ > 0 ” for some real constant c. Together with (5.2) we shall describe two other classes of functions highlighting a possible concept of sublogarithmic variation which fits, e.g., to iterated logarithms.

Definition 5.1. Let f be a measurable function defined on a neighborhood of + .

(I) We say that f is quasi-logarithmically varying at + if:

f ( λ x ) = f ( x ) + O ( 1 ) , x + , fixed λ > 0. (5.3)

(II) We say that f is logarithmically varying at + if:

f ( λ x ) = f ( x ) + c log λ + o ( 1 ) , x + , fixed λ > 0 and some constant c \ { 0 } . (5.4)

(III) We say that f is hypo [ sub]-logarithmically varying at + if:

f ( λ x ) = f ( x ) + o ( 1 ) , x + , fixed λ > 0. (5.5)

Equations (5.3) and (5.4) have trivial solutions, namely all bounded functions and all functions convergent as x + respectively satisfy (5.3) and (5.4); and obviously the behaviors of such functions have nothing to partake of the intuitive meanings associated with “logarithmic variation”. For this reason someone might prefer to use the locutions of “logarithmically varying” and “hypo-logarithmically varying” exclusively for functions which, besides satisfying the appropriate above-specified equation, enjoy additional properties such as: strict positivity, or monotonicity, or divergence to + as x + , or the last two properties. But this is a matter of agreement. Now we collect the essential properties of the three classes using the shortened locution “ f differentiable” to mean that:

either “ f is everywhere differentiable on some interval [ T , + )

or “ f is absolutely continuous on [ T , + ) ”,

with the agreement that an asymptotic relation involving f , say “ f ( x ) ϕ ( x ) , x + ” for an absolutely continuous f is to be meant as “ x + , x [ T , + ) \ N ” where N is a suitable set of measure zero.

Three admissible meanings of the locution “ f monotonic” have been highlighted at the outset of the preceding section and the proofs involving this property may be done using the integral mean-value theorem which applies to each of the three cases.

Theorem 5.1. (I) If f is a measurable function defined on a neighborhood of + satisfying Equation (5.3) then the following two properties hold true:

1) The asympyotic relation (5.3) holds true uniformly with respect to the parameter λ on each compact subset of ] 0, + ) and this implies that f satisfies the more general asymptotic functional equation:

f ( x I ( x ) ) = f ( x ) + O ( 1 ) , x + , function I such that 0 < c 1 I ( x ) c 2 < + ; (5.6)

and in particular:

( f ( λ x + o ( x ) ) = f ( x ) + O ( 1 ) , x + , uniformly for λ varying on a compact subset of ] 0, + ) . (5.7)

2) f satisfies the estimate:

f ( x ) = O ( log x ) , x + . (5.8)

(II) If f is a differentiable function then condition f ( x ) = O ( x 1 ) , x + implies that f satisfies Equation (5.3).

(III) If f is monotonic then (5.3) is satisfied iff

f ( x ) = O ( x 1 ) , x + . (5.9)

Theorem 5.2. (I) If f is a measurable function defined on a neighborhood of + satisfying Equation (5.4) then it obviously satisfies (5.3), hence (5.7); moreover:

( f ( λ x + o ( x ) ) = f ( x ) + c log λ + o ( 1 ) , x + , uniformly for λ varying on a compact subset of ] 0 , + ) ; (5.10)

and the growth-order of f is:

f ( x ) c log x , x + . (5.11)

(II) If f is a differentiable function then condition f ( x ) c x 1 , x + implies that f satisfies Equation (5.4).

(III) If f is monotonic then (5.4) is satisfied iff

f ( x ) c x 1 , x + . (5.12)

Theorem 5.3. If f is a measurable function defined on a neighborhood of + satisfying Equation (5.5) then a word-for-word restatement of Theorem 5.1 holds true with the symbol “O” replaced everywhere by “o”.

Proof of the theorems. The fundamental results about the uniformity of equations (5.3)-(5.4)-(5.5) with respect to λ may be found in the monograph ( [4]; Corollaries 3.1.8a and 3.1.8c, case g 1 , p. 133) for equations (5.3) and (5.5); whereas (5.4) is a special case of (5.3). From such results the following useful representations are derived ( [4]; Theorem 3.6.1, p. 152):

f ( x ) = C + T 0 + η ( x ) + T 0 x ξ ( t ) t 1 d t , x T 0 , (5.13)

where C , T 0 are suitable constants and the measurable functions η , ξ are such that:

η ( x ) , ξ ( x ) = ( O ( 1 ) if f satisfies ( 5.3 ) , o ( 1 ) if f satisfies ( 5.5 ) ; (5.14)

( η ( x ) = O ( 1 ) , ξ ( x ) = c + o ( 1 ) , if f satisfies (5.4). (5.15)

The estimates in (5.14) are explicitly stated in ( [4]; Th. 3.6.1, p. 152) whereas those in (5.15) are checked at once by inspecting a formula in ( [4]; formula (3.6.2), p. 152). From (5.13) the relations in (5.8) and (5.11) and the corresponding one in Theorem 5.3 are easily inferred. So the statements in part (I) of each theorem are proved, and the statements in parts (II) trivially follow from the integral representation:

f ( λ x ) f ( x ) = x λ x f ( t ) d t . (5.16)

For parts (III) in the theorems we must prove the “only if” inferences noticing that, changing f in - f if necessary, we may always suppose f 0 . Relation (5.9) follows from:

( f non-decreasing [ f ( 2 x ) f ( x ) ] = x 2 x f x f ( x ) ; f non-increasing [ f ( x / 2 ) f ( x ) ] = x / 2 x f x 2 f ( x ) . (5.17)

The very same calculations also give the estimate with “o” for Theorem 5.3. Relation in (5.12) requires less immediate calculations. For f non-decreasing and each λ > 1 we have:

f ( x ) 1 ( λ 1 ) x x λ x f f ( λ x ) f ( x ) ( λ 1 ) x f ( λ x ) , (5.18)

whence:

x f ( x ) c log λ + o ( 1 ) λ 1 λ 1 ( λ x ) f ( λ x ) ; (5.19)

and letting x + :

lim ¯ x + x f ( x ) c log λ λ 1 lim _ x + λ 1 ( λ x ) f ( λ x ) = λ 1 lim _ x + x f ( x ) , λ > 1. (5.20)

As λ 1 + we get relation (5.12). For f non-increasing and λ > 1 the inequalities in (5.19) are reversed and an analogous reasoning may be done. We also get the estimate with “o” for Theorem 5.3.

And now what can be said about the concept of higher-order logarithmic variation? Higher-order regular variation is defined by imposing on each derivative of order not less than 2 an asymptotic behavior consistent with that of the first derivative according to a preliminary result: ( [1]; Prop. 2.6, p. 796, and Def. 3.1, p. 798); the foregoing Theorems 5.1-5.3 justify the following:

Definition 5.2. If f is n-times differentiable on an interval [ T , + ) , n 1 , in the sense that f ( n 1 ) is either absolutely continuous or everywhere differentiable, then one may use the following locutions:

( f is quasi-logarithmically varying at + of order n f ( k ) ( x ) = O ( x k ) , x + , 1 k n ; f is logarithmically varying at + of order n f ( k ) ( x ) c D k ( log x ) , x + ; c 0, 0 k n ; ( D 0 h ( x ) : = h ( x ) ) ; f is hypo-logarithmically varying at + of order n f ( k ) ( x ) = o ( x k ) , x + , 1 k n . (5.21)

Remarks. 1) Some people would like to add to the above definition a condition such as “strict positivity or monotonicity or divergence” for the sole function f (and not for its derivatives!) to adhere more consistently to the intuitive notion of logarithmic variation; but this is a matter of agreement as noticed above.

2) The asymptotic estimate (5.9) obviously implies (5.8) but the converse fails regardless of any monotonicity restriction; in fact for, say, a C 1 function we have:

( f ( x ) log x g ( x ) g ( x ) = O ( 1 ) f ( x ) x 1 g ( x ) + log x g ( x ) = O ( x 1 ) g ( x ) = O ( ( x log x ) 1 ) . (5.22a)

And in the case “ g ( x ) = c + o ( 1 ) , c , the characteristic condition for f to satisfy the corresponding asymptotic relation “ f ( x ) = c x 1 + o ( x 1 ) ” is “ g ( x ) = o ( ( x log x ) 1 ) ”. These remarks show that for the validity of the inference “(5.8) (5.9)” the right additional condition (beside differentiability) is the following restriction on the order of growth of the derivative:

( f ( x ) / log x ) = O ( ( x log x ) 1 ) . (5.22b)

From (5.22a) and Theorem 5.1 it follows that the solutions of (5.3) which are ultimately concave or convex satisfy the three estimates: of

f ( x ) = O ( log x ) ; f ( x ) = O ( x 1 ) ; ( f ( x ) / log x ) = O ( ( x log x ) 1 ) , x + . (5.23)

Some examples. 1) The standard ones. The following functions are hypo-logarithmically varying at + of a non-trivial type:

( ( log x ) δ , 0 < δ < 1 ; 2 k n ( l k ( x ) ) δ k , δ k ; ( log x ) δ 2 k n ( l k ( x ) ) δ k , 0 < δ < 1 , δ k (5.24)

The first of the foregoing function shows the obvious fact that no one of the three classes in Definition 5.1 is closed under multiplication.

The following examples concern various types of compositions.

2) Logarithm of a regularly-varying function:

( f R α ( + ) , α log f logarithmically varying at + , as log ( f ( λ x ) ) = log ( λ α f ( x ) [ 1 + o ( 1 ) ] ) = log ( f ( x ) ) + α log λ + o ( 1 ) , x + . (5.25)

3) A regularly-varying function of the logarithm. The general result is:

( g quasi-logarithmically varying at + , g ( + ) = + , f asymptotically sublinear at + , f g hypo-logarithmically varying at + . (5.26)

In fact the assumptions mean that g satisfies “ g ( λ x ) = g ( x ) + O ( 1 ) ” and f satisfies “ f ( x + O ( 1 ) ) = f ( x ) + o ( 1 ) ” ( [1]; Prop. 5.2-(I), p. 814), hence:

f ( g ( λ x ) ) = f ( g ( x ) + O ( 1 ) ) = f ( g ( x ) ) + o ( 1 ) , x + . (5.27)

The most meaningful contingency for f in (5.27) is:

( f differentiable with f ( x ) = o ( 1 ) , x + ; in particular : f R δ ( + ) , δ < 1 , so generalizing the first example in ( 5.24 ) . (5.28)

The case “ f R 1 ( + ) “ needs restrictions:

( g hypo-logarithmically varying at + , g ( + ) = + , f asymptotically uniformly continuous at + , f g hypo-logarithmically varying at + ; (5.29)

whose proof is as above, noticing that now “ g ( λ x ) = g ( x ) + o ( 1 ) ” and f satisfies, by definition, “ f ( x + o ( 1 ) ) = f ( x ) + o ( 1 ) ” ( [1]; Prop. 5.2-(I), p. 814); and a sufficient condition for such an f is “ f ( x ) = O ( 1 ) ”.

4) A slowly-varying function of the logarithm. Here is a result useful in inverting an hyper-exponential function:

Proposition 5.4. (I) If l { R 0 ( + ) oforder n } and l ( k ) R k ( + ) , 1 k n 1 , then the function L : = l log has the following properties:

( L { R 0 ( + ) of order n } ; L ( k ) R k ( + ) , 1 k n 1 ; L ( k ) ( x ) ( 1 ) k 1 _ x k l ( log x ) ( 1 ) k 1 _ x 1 k L ( x ) , x + , 1 k n ; L ( k )