A Theorem of Galambos-Bojanic-Seneta Typedownloads.hindawi.com/journals/aaa/2009/360794.pdf · A...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2009, Article ID 360794, 6 pagesdoi:10.1155/2009/360794

Research ArticleA Theorem of Galambos-Bojanic-Seneta Type

Dragan Djurcic1 and Aleksandar Torgasev2

1 Technical Faculty, University of Kragujevac, Svetog Save 65, Cacak 32000, Serbia2 Faculty of Mathematics, University of Belgrade, Studentski trg 16a, Belgrade 11000, Serbia

Correspondence should be addressed to Dragan Djurcic, [email protected]

Received 9 January 2009; Accepted 29 March 2009

Recommended by Stephen Clark

In the theorems of Galambos-Bojanic-Seneta’s type, the asymptotic behavior of the functionsc[x], x ≥ 1, for x → +∞, is investigated by the asymptotic behavior of the given sequence ofpositive numbers (cn), as n → +∞ and vice versa. The main result of this paper is one theoremof such a type for sequences of positive numbers (cn)which satisfy an asymptotic condition of theKaramata type limn→∞ c[λn]/cn > 1, for λ > 1.

Copyright q 2009 D. Djurcic and A. Torgasev. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

1. Introduction

A function f : [a,+∞) �→ (0,+∞) (a > 0) is called O-regularly varying in the sense of Karamata(see [1]) if it is measurable and if for every λ > 0,

kf(λ) := limx→+∞

f(λx)f(x)

< +∞. (1.1)

Function kf(λ) (λ > 0) is called the index function of f , and ORVf is the class of allO-regularly varying functions defined on some interval [a,+∞).

A function f ∈ ORVf is called O-regularly varying in the Schmidt sense (see [2, 3]) if

limλ→ 1

kf(λ) = 1. (1.2)

O-regularly varying functions in the Schmidt sense form the functional class IRVf

and IRVf � ORVf (see [3]). They represent an important object in the analysis of divergentprocesses (see [4–9]). In particular, we have that the class RVf of regularly varying functions

2 Abstract and Applied Analysis

in the Karamata sense satisfies RVf � IRVf (see [3], and some of its applications can be foundin [10]).

A function f ∈ IRVf is called regularly varying in Karamata sense if kf(λ) = λρ for everyλ > 0 and a fixed ρ ∈ R. If ρ = 0, then f is called slowly varying in the Karamata sense , and allsuch functions form the class SVf . We have that SVf � RVf (see [10]).

A sequence of positive numbers (cn) is called O-regularly varying in the Karamata sense(i.e., it belongs to the class ORVs), if

kc(λ) = limn→+∞

c[λn]

cn< +∞, (1.3)

for every λ > 0.A sequence (cn) ∈ ORVs is calledO-regularly varying in the Schmidt sense (i.e., it belongs

to the class IRVs), if

limλ→ 1

kc(λ) = 1. (1.4)

The classes of sequences ORVs and IRVs have an important place in the qualitativeanalysis of sequential divergent processes (see, e.g., [11–14]). Asymptotic properties ofsequences (1.3) and (1.4) are very important in the Theory of Tauberian theorems (see [7, 15]).

The class of regularly varying sequences in the Karamata sense RVs and similarly theclass of slowly varying sequences in the Karamata sense SVs are defined analogously to theclasses RVf and SVf . They are fundamenatal in the theory of sequentional regular variabilityin general (see [16]).

Next, let (cn) be a strictly increasing, unbounded sequence of positive numbers. Then

δc(x) = max{n ∈ N | cn ≤ x}, (1.5)

for x ≥ c1, is the numerical function of the sequence (cn) (see, e.g., [17]).In the sequel, let ∼ be the strong asymptotic equivalence of sequences and functions,

and let (pn) be the sequence of prime numbers in the increasing order. Since pn ∼ n ln n,n → +∞ ((pn) ∈ IRVs) and since δp(x) ∼ (x/ ln x), x → +∞, (δp ∈ IRVf) (see, e.g., [17]),the next question seems to be natural:

what is the largest proper subcclass of the class of all strictly increasing, unboundedsequences from IRVs, such that the numerical function of every one of its elements belongsto IRVf?

The next example shows that this question has some sense.

Example 1.1. Define c1 = ln 2/2 and cn = ln n, for n ≥ 2. Then (cn) is a strictly increasing,unbounded sequence of positive numbers. Since ln x, x ≥ 2 belongs to the functional classSVf (see, e.g., [10]), by a result from [18], we have that (cn) ∈ SVs. Hence, (cn) ∈ IRVs. Next,since δc(x) ∼ h−1(x), x → +∞, where h(x), x ≥ 1, is continuous and strictly increasing, andh(n) = cn (n ∈ N) (see, e.g., [17]), we can assume that h(x) = ln x for x ≥ 2, while forx ∈ [1, 2) we can suppose that h is linear and continuous on [1, 2] such that h(1) = ln 2/2.

Abstract and Applied Analysis 3

Therefore, δc(x) ∼ ex, as x → +∞, so that δc belongs to de Haan class of rapidly varyingfunctions with index +∞ (the class R∞,f) (see, e.g., [19]). Hence, if λ > 1 we have

limx→+∞

δc(λx)δc(x)

= +∞, (1.6)

so that δc does not belong to IRVf .

Knowing of asymptotic characteristics of a considered sequence and of its numericalfunction can be of a great importance in many constructions of the asymptotic analysis (see,e.g., [17]).

Next, we say that a function f : [a,+∞) �→ (0,+∞), a > 0, belongs to the class ARVf ifit is measurable and for every λ > 1 we have

kf(λ) = limx→+∞

f(λx)f(x)

> 1. (1.7)

The function kf(λ), λ > 0, is the auxiliary index function of the function f(x), x ≥ a.Condition (1.7) is equivalent with assumption that there exists an x0 = x0(λ) ≥ a and

c(λ) > 1 for λ > 1, so that for every λ > 1 and every x ≥ x0 it holds

f(λx) ≥ c(λ) · f(x). (1.8)

The class ARVf contains (as proper subclasses) the class of all regularly varyingfunctions in the Karamata sense whose index of variability is positive as well as the classof all rapidly varying functions in de Haan sense whose index of variabilty is +∞, but it doesnot contain any slowly varying function in the Kararamata sense.

We also have that ARVf ∩ IRVf /=∅ and ARVfΔIRVf /=∅. Besides, the class ARVf

considered in the space of the so-called ϕ-functions (see, e.g., [8]) is also an essential objectof the asymptotic and the functional analysis (see, e.g., [20]).

Next, let ARVs be the class of all positive numbers (cn) such that for every λ > 1 wehave

kc(λ) = limn→+∞

c[λn]

cn> 1. (1.9)

The function kc(λ), λ > 0, is called the auxiliary index function of the sequence (cn).The above condition is equivalent with fact that there is an n0 = n0(λ) ∈ N and a

function c(λ) > 1, λ > 1, such that for every λ > 1 and for every n ≥ n0 we have

c[λn] ≥ c(λ) · cn. (1.10)

The class ARVs contains (as proper subclasses) the class of all regularly varyingsequences in the Karamata sense whose index of variability is positive as well as the classof all rapidly varying sequences in de Haan sense whose index is +∞, but does not containany slowly varying sequence in the Karamata sense (see [21, 22]).

We also have that ARVs ∩ IRVs /=∅ and ARVsΔIRVs /=∅.


2. Main Results

The next theorem is a theorem of Galambos-Bojanic-Seneta type (see [16, 18]) for classesARVs and ARVf . The analogous theorems for regularly varying sequences and functionsin the Karamata sense, O-regularly varying sequences and functions in the Karamata sense,sequences from the class IRVs and functions from the class IRVf , rapidly varying sequencesand functions in de Haan sense with index +∞, the Seneta sequences and functions (see, e.g.,[23]) can be found, respectively, in [13, 16, 24–27].

Theorem 2.1. Let (cn) be a sequence of positive numbers. Then the next assertions are equivalent asfollows:

(a) (cn) ∈ ARVs,

(b) f(x) = c[x], x ≥ 1, belongs to the class ARVf .

Proof. (a) ⇒ (b) Let (cn) be a sequence of positive numbers and assume that (cn) ∈ ARVs,thus that limn→+∞(c[λn]/cn) > 1 for every λ > 1. If λ > 1 is arbitrary fixed number, thenkc(α) > 1 for every α ∈ (1, λ). For arbitrary α ∈ (1, λ) define nα ∈ N in the following way:nα = 1 if c[αn]/cn > 1 for every n ∈ N, and nα = 1 +max{n ∈ N | c[αn]/cn ≤ 1} else. One caneasily see that 1 ≤ nα < +∞ for every considered α.

Next, define a sequence of sets (Ak) by Ak = {α ∈ (1, λ) | nα > k} (k ∈ N). Then thissequences is nonincreasing, thus Ak+1 ⊆ Ak (k ∈N) and

⋂∞k=1Ak = ∅. We shall show that not

all subsetsAk (k ∈N) are dense in (1, λ). If α ∈ Ak for a fixed k ∈N, then c[(nα−1)α]/cnα−1 ≤ 1,and there is a δα > 0 such that c[(nα−1)t]/cnα−1 = c[(nα−1)α]/cnα−1 ≤ 1, for every t ∈ [α, α + δα) �

(1, λ). Hence, every t ∈ (α, α + δα) belongs to Ak, since nt ≥ (nα − 1) + 1 > k. This givesthat (α, α + δα) ⊆ Ak if α ∈ Ak. Assuming now that a set Ak is dense in (1, λ), we get thatthe set IntAk is also dense in (1, λ). If else, we assume that all sets Ak (k ∈ N) are dense in(1, λ), we find that (IntAk) is a sequence of open dense subsets of the set (1, λ) of the secondcategory. Then we get that the set

⋂∞k=1Ak is dense in (1, λ), so it must be nonempty, which is

a contradiction. Hence, we conclude that there is an n0 ∈ N, so that the set An0 is not densein (1, λ). Hence, there is an intervals [A,B] � (1, λ) (A < B) such that [A,B] ⊆ (1, λ) \An0 ={α ∈ (1, λ) | nα ≤ n0}.

Therefore, for every α ∈ [A,B]we have nα ≤ n0. Hence, for every n ≥ n0 ≥ nα and everyα ∈ [A,B] we have c[αn]/cn > 1. Consequently, for any λ ∈ (1,+∞) and all sufficiently largex ≥ x0 we have that c[λx]/c[x] = (c[t[η[x]]]/c[η[x]]) · (c[η[x]]/c[x]), where t = t(x) ∈ [A,B] andη = 2λ/(A + B). Since η > 1, we get limx→+∞c[λx]/c[x] ≥ kc(η) > 1, so that f(x) = c[x] (x ≥ 1)belongs to the class ARVf .

Since (b)⇒ (a) is immediate, we completed the proof.

The above theorem provides (analogously, as in cases given before Theorem 2.1) aunique development of the theory of sequences from the class ARVs and theory of thefunctions from the class ARVf . Thus, Theorem 2.1 can be used to interpret all asymptoticbehaviors of functions from the class ARVf (some of them are given in [28]) as behavior ofsequences from the class ARVs and vice versa.

Corollary 2.2. Let (cn) be a strictly increasing unbounded sequence of positive numbers. Then,

(a) (cn) ∈ ARVs if and only if δc(x) (x ≥ c1) ∈ IRVf ;

(b) (cn) ∈ IRVs if and only if δc(x) (x ≥ c1) ∈ ARVf .

Abstract and Applied Analysis 5

Proof. (a) Let (cn) be a strictly increasing unbounded sequence of positive numbers, andassume that (cn) ∈ ARVs. Then by Theorem 2.1, f(x) = c[x], x ≥ 1, belongs to ARVf . f isnondecreasing and unbounded for x ≥ 1. Let f←(x) = inf{y ≥ 1 | f(y) > x}, x ≥ c1, bethe generalized inverse (see [1]) of f . It is correctly defined nondecreasing and unboundedfunction for x ≥ c1. It is also stepwise and right continuous. We also have that δc(x) =f←(x) − 1 for x ≥ c1.

According to [22] we have that function f←(x), x ≥ c1, belongs to the class IRVf .Since f← is nondecreasing and unbounded, we get limx→+∞(δc(x)/f←(x)) = 1, so that

δc(x), x ≥ c1, belongs to IRVf .Next, let (cn) be a strictly increasing unbounded sequence of positive numbers, and let

δc(x), x ≥ c1, belong to IRVf . Besides, let f(x) = c[x], x ≥ 1. Since f←(x) = δc(x) + 1 for x ≥ c1,we find that f← ∈ IRVf . According to [28] we have that function f(x), x ≥ 1, belongs to theclass ARVf . So by Theorem 2.1 we get that (cn) ∈ ARVc.

(b) Now, assume that (cn) is a strictly increasing unbounded sequence of positivenumbers and (cn) ∈ IRVs. Then by [13], f(x) = c[x], x ≥ 1, belongs to IRVf . Analogouslyto (a), then δc(x) = f←(x) − 1, x ≥ c1. According to [29] (or [28]) we have that functionf←(x), x ≥ c1, belongs to the class ARVf , and consequently δc ∈ ARVf .

Next, let (cn) be a strictly increasing unbounded sequence of positive numbers, andassume that δc ∈ ARVf . Besides, let f(x) = c[x], x ≥ 1. Since f←(x) = δc(x) + 1, for x ≥ c1,then f← ∈ ARVf . According to [29] (or [28]) we have that function f(x), x ≥ 1, belongs tothe class IRVf . According to [13] the sequence (cn) (cn = f(n), n ∈N) belongs to IRVs.

Let K∗,ic,s be the class of all strictly increasing unbounded sequences from the classIRVs ∩ ARVs (see [8]). This class contains (as a proper subclass) all strictly increasingunbounded regularly varying sequences in the Karamata sense whose index of variabilityis positive, and it does not contain any sequence from the class SVs, nor from the class R∞,s.

The next statement gives the answer to the question from the introduction of thispaper. It is a corollary of Corollary 2.2 (and, indirectly, of Theorem 2.1).

Corollary 2.3. The class K∗,ic,s is the largest proper subclass of the class of strictly increasingunbounded sequences from the class IRVs, such that the numerical function of any its element belongsto the class IRVf .

Proof. Let (cn) be a strictly increasing unbounded sequence of positive numbers from the classIRVs ∩ARVs. Then by Corollaries 2.2(a) and 2.2(b), δc ∈ IRVf ∩ARVf , thus δc ∈ IRVf . Next,assume that (cn) is a strictly increasing unbounded sequence of positive numbers from theclass IRVs \ARVs. Then by Corollary 2.2(b), δc ∈ ARVf , and by Corollary 2.2(a) δc /∈ IRVf .

This completes the proof.

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