If this is the case then the following integral representation holds true

(3.8)

for a suitable polynomial, the same as above, and a suitable number, the same as in (3.6).

We expressed relations in (3.7) by saying that the asymptotic expansion

(3.9)

is formally differentiable n times in the “strong sense” because in the same paper we characterized another weaker set of differentiated expansions, ([1] ; Th. 3.1, p. 173), which we shall not presently use.

Proposition 3.2. If and is convex of order on―which is equivalent to the property that is increasing thereon―then: f has a “polynomial asymptotic expansion at +∞”, i.e. it satisfies a relation of type

(3.10)

iff its nth-order contact indicatrix is bounded (hence, by monotonicity, condition (3.6) holds true). If this is the case then we also have the properties in Proposition 3.1, hence the expansion (3.10) automatically implies its formal differentiability n times in the strong sense.

Now we give analogues of the two foregoing propositions with condition (3.6) replaced by the weaker condition; strong differentiability will be granted times and the validity of an expansion (3.10) will be characterized for a class of functions larger than nth-order convexity.

Theorem 3.3. For, the following are equivalent properties:

1) All the functions

(3.11)

2) The single function

(3.12)

3) There exists a polynomial such that

(3.13)

If this is the case then and the following integral representation holds true:

(3.14)

In the elementary case n = 1 the result is:

(3.15)

Notice that the representation of inferred from (3.14) contains the quantity hence, by the example in (2.13), no information on the growth-order of may be obtained in the context of Theorem 3.3, generally speaking.

For a characterization similar to that in (3.15) holds true under a restriction on the sign of and we have the following analogue of Proposition 3.2.

Theorem 3.4. Let and let satisfy a one-sided boundedness condition:

(3.16)

Then an expansion (3.10) holds true iff. If this is the case then, according to Theorem 3.3, the expansion (3.10) is formally differentiable times in the strong sense.

We exhibit an example for the case and a counterexample for the case; they seem to be just the same because in both expansions the remainder is exactly the same quantity but a striking difference appears in the behaviors of and.

Example for the case:

(3.17)

Here is bounded and admits of asymptotic mean but has no limit at +∞; accordingly the expansion is not formally differentiable in the strong sense though the differentiated expansions of any order satisfy the remarkable asymptotic estimates in (3.17).

Counterexample for the case:

(3.18)

Here
is unbounded both from below and from above and admits of no asymptotic mean; notwithstanding,
an asymptotic expansion
holds true. Hence the equivalence stated in Theorem 3.4 may fail without the restriction
in (3.16). According to Theorem 3.3 the expansion of f_{2} is not formally
differentiable once in the strong sense.

In the elementary case in (3.15) condition is explicitly defined in Giblin ([24] ; p. 279) as the “bounded distance condition” and it is easily checked that it is equivalent to a pair of relations

(3.19)

it is the further condition of existence of asymptotic mean that changes the first relation in (3.19) into an asymptotic straight line.

4. Two-Term Asymptotic Expansions and Asymptotic Means

In this section we give an exhaustive list of results concerning the role of asymptotic mean in the theory of two-term asymptotic expansions involving comparison functions admitting of indexes of variation at +∞. We first report a result from [4] .

Preliminary notations and formulas ([4] ; p. 255). As usual we say that two functions
f, g (as well as their graphs) have a first-order contact at a point t_{0}
if
and
provided that f, g are defined on a neighborhood of t_{0} and the involved
derivatives exist as finite numbers.

Let now,
be
two real-valued functions differentiable on an interval I such that their Wronskian
never vanishes on I and let f be differentiable on I. Then for each
there exists a unique function in the family
having a first-order contact with f at t_{0}. Denoting this function by
we have

(4.1)

where

(4.2)

If
then
on I for any chosen t_{0}. The function

(4.3)

will be called the contact indicatrix of order one of the function f at the point t with respect to the family and the straight line.

represents the ordinate of the point of intersection between the vertical line and the curve where t is thought of as fixed. The assumption on implies that and do not vanish simultaneously hence is a nontrivial linear combination of,. It may happen that, for some choices of T, coincides with or, a constant factor apart, according as or.

Using (4.2) may be represented as

(4.4)

where we have put

(4.5)

Proposition 4.1. (Characterization of a two-term asymptotic expansion: [4] , Th. 4.4, p. 258). Assumptions:

(4.6)

(4.7)

For a function the following are equivalent properties:

1) It holds true an asymptotic expansion

(4.8)

2) There exists a finite limit

(4.9)

3) There exists a finite limit

(4.10)

If this is the case we have the following two representations:

(4.11)

(4.12)

The validity of (4.8) may be expressed by the geometric locution: “the graph of f admits of the curve as an asymptotic curve in the family, as.”

Notice that in the cited reference condition (4.10) is written in the form

(4.13)

however (4.5) implies

(4.14)

and (4.10) follows.

The two limits in (4.9), (4.10) are of the type studied in §2 and a direct application of Theorem 2.4 gives the following results.

Theorem 4.2. In assumptions (4.6)-(4.7) let it be:;.

(I) (Regularly-varying comparison functions). If

(4.15)

then the following three properties are equivalent:

(4.16)

(4.17)

(4.18)

(II) (Slowly-varying comparison functions). If

(4.19)

then each condition (4.17) or (4.18) implies an expansion (4.16).

(III) (Rapidly-varying comparison functions). Put and suppose that:

(4.20)

then an expansion (4.16) implies both conditions (4.17)-(4.18).

Under the stated assumptions for the validity of part (I) the equivalence “(4.16) Û (4.18)” admits of the following geometric reformulation:

“The graph of f admits of an asymptotic curve in the family, as, iff the contact indicatrix of order one of the function f with respect to has an asymptotic mean at +∞”.

Notice that this result for two-term expansions requires no restrictions on the signs of,.

5. Proofs

Proof of Lemma 2.3. By hypothesis the following two limits exist in:

(5.1)

We now evaluate by L’Hospital’s rule first noticing that: implies, whereas for it is and the first limit in (5.1) implies. In both cases the rule may be applied and

(5.2)

It remains the case which implies and this condition leads to excluding the following contingencies for the indicated reasons:

1)

2) (by L’Hospital’s rule)

which is a positive real number; hence which would imply.

3) and this would imply, by L’Hospital’s rule as in (5.2):.

4) The case must be treated in a different way. A basic property of our class of functions, directly inferred from the limits in (5.1), claims the validity of the following asymptotic estimates:

(5.3)

Now in our present proof we have and, hence

and there are two a-priori contingencies about the integral. Its divergence would imply which cannot be; in the other case we would have

(5.4)

which contradicts the second relation in (5.3). Notice that the procedure used to prove this last case works for any as well.

The last assertion in the statement of Lemma 2.3, namely “it cannot be”, follows from the calculations in 2): implies, but in this case (5.2) shows, a contradiction. W

Proof of Theorem 2.4. (I) We make explicit the assumptions writing:

(5.5)

which in turn imply the following relations to be used in the sequel:

(5.6)

(5.7)

(5.8)

First part: (2.17) Þ (2.1). If we put

(5.9)

then, by (2.17), we may write

(5.10)

From (5.9) and (2.17):

(5.11)

(5.12)

(5.13)

Using (5.11) and (5.13) in the left side of (5.10) we get, i.e. (2.1).

Second part: (2.1) Þ (2.17). First step: convergence of Consider the identity

(5.14)

and estimate the behavior of as. From (2.1) and 5.8) we get:

(5.15)

(5.16)

As concerns we have:

(5.17)

from whence and (2.1) we get:

(5.18)

As, we obtain the convergence of hence, by (5.14), of.

Second step: asymptotic behavior of. By (5.16) and (5.18) we may integrate by parts as follows:

(5.19)

which is (2.17) with.

(II) From the first assumption in (2.27) we infer:

(5.20)

and from (5.17):

(5.21)

Now we retrace all steps in the second part of the proof of part (I) checking the validity of the corresponding formulas for. Instead of the first relation in (5.16) we have:

(5.22)

and, instead of (5.18):

(5.23)

The convergence of follows as above. And using the same integration by parts as in (5.19) we get the same final relation.

(III) Let us first show that the three conditions in (2.28) imply that both
are rapidly-varying at +∞. Conditions in (2.28)_{1,2} are equivalent to,
and (2.28)_{3} is equivalent to

(5.24)

which implies, by (2.28)_{1},
ultimately;
so we have:

Now we retrace all steps in the first part of the proof of part (I) and again use decomposition (5.10); instead of (5.11) we get:

(5.25)

and instead of (5.12) we get, using (5.24):

(5.26)

whence

(5.27)

From (5.25), (5.26), (5.27) we get (2.1) with. W

Proof of Proposition 2.6. Integration by parts gives:

(5.28)

whence our claim follows dividing both sides by x. W

Proof of Theorem 3.3. Let us assume (3.12) and start from the integral representation ([1] ; formula (6.3), p. 185):

(5.29)

which for reads:

(5.30)

From (5.30) the elementary equivalence in (3.14) easily follows, hence we suppose. If (3.12) holds true and we apply the asymptotic relation in (2.29) to we get:

(5.31)

and the last relation, when replaced into (5.29), yields:

(5.32)

But the first relation in (5.31) implies that the iterated improper integral converges and we get a representation of type:

(5.33)

together with the expansion:

(5.34)

having used one of the following elementary identities (to be used again):

(5.35)

To prove the formal differentiabilty we put:

(5.36)

and from (5.31) we infer relations:

(5.37)

Calling the last sum on the right in (5.34), which differ by a constant from the sum on the right in (5.33), and applying Leibniz's rule to (5.33) we get:

(5.38)

The expressions of and involve and its derivative:

(5.39)

So far we have proved that (3.12) implies relations in (3.13) for, without any information on, and, for the time being, is a non-better specified polynomial of degree. To prove (3.11) we estimate the behavior, as, of for using its known expression in terms of f, ([1] ; formula (2.6), p. 168):

(5.40)

as the first sum is nothing but the expression of the coefficient of the power in the polynomial, i.e.,

By (2.34) the function has asymptotic mean “zero” and the same is true for a term; so the sum of the last three terms above represents a function with asymptotic mean equalling. We have proved that “2) Þ 1) Ù 3)”. It remains to show “3) Þ 2)”. First step. Let us first evaluate from representation (5.29); putting

(5.41)

we get:

(5.42)

Now we start as in (5.40) from the expression of:

(5.43)

whence we get

(5.44)

which implies the convergence of the improper integral; and we can rewrite representation (5.29) in the form:

(5.45)

Comparing (5.45) and the assumed relation we infer that the two polynomials and the sum appearing in (5.45) have the same leading coefficient:. Now we do calculations just like those from (5.41) to (5.43) but starting from representation (5.45) and paying attention to the signs, so getting:

(5.46)

(5.47)

having used the identity, ([1] ; Lemma 2.2, p. 169). From (5.47) we infer

(5.48)

which, by (2.29), implies and. W

Proof of Theorem 3.4. The only thing to be proved is that an expansion (3.10) plus condition (3.16) imply. We first show that it is enough to prove our claim with (3.16) replaced by the condition of one-signedness:

(5.49)

In fact it is known, ([1] ; Lemma 2.2, p. 169), that: iff f is a polynomial of type

(5.50)

Let now g be any function, , let p be a polynomial of type (5.50) and define: . With an obvious meaning of the symbol we have:; hence:

(5.51)

It follows that any result on formal differentiability of a polynomial asymptotic expansion involving g admits of a literal transposition to a polynomial asymptotic expansion involving f. Our assumption are now: expansion (3.10) and one-signedness of, and the proof (which we make explicit here) is a word-for-word repetition of that in ([1] ; Proof of Th. 4.2, pp. 193-195) with a slight modification at the conclusive passage. From representation (5.29) we infer

(5.52)

and, by (3.10), the following limit:

(5.53)

For (3.10) reduces to and (5.53) is “convergent”. Hence representation (5.29) can be rewritten in the form

and (3.10) implies that “exists in” which is equivalent to.

For we apply L’Hospital’s rule times to the limit in (5.53) so getting the limit:

(5.54)

By the one-signedness of this last limit exists in the extended real line, hence it must be a finite number. This means the convergence of and representation (5.29) can be rewritten as:

(5.55)

The last relation implies that coincides with the in (3.10) and we get:

(5.56)

By the above argument involving L’Hospital’s rule we arrive at the convergence of the iterated integral

. An iteration of the procedure yields condition

(5.57)

which implies representation

(5.58)

where the coefficients are those in (3.10). From (5.58) we infer that

(5.59)

and applications of L’Hospital’s rule times yields the limit

(5.60)

which, by (2.29), is equivalent to. W

In passing notice that the last calculations and (5.34) prove that:

For a given function and g one-signed the following equivalence holds true:

(5.61)

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