# A Theorem of Galambos-Bojanić-Seneta Type

Authors: Dragan Djurčić; Aleksandar Torgašev
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

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


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## Body

## 1. Introduction

A functionf:[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, (1.1)k¯f(λ):=lim¯x→+∞f(λx)f(x)<+∞.Functionk¯f(λ)(λ>0) is called the index function of f, and ORVf is the class of
all 𝒪-regularly
varying functions defined on some interval [a,+∞).A functionf∈ORVf is called 𝒪-regularly
varying in the Schmidt sense (see
[2, 3]) if(1.2)lim¯λ→1k¯f(λ)=1.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 divergent processes (see [4–9]). In particular, we have
that the class RVf of regularly
varying functions in the Karamata sense satisfies RVf⊊IRVf (see [3], and some of its
applications can be found in [10]).A functionf∈IRVf is called regularly varying in Karamata sense if k¯f(λ)=λρ for every λ>0 and a fixed ρ∈ℝ. If ρ=0, then f is called slowly varying in the Karamata sense , and
all such 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(1.3)k¯c(λ)=lim¯n→+∞c[λn]cn<+∞,for every λ>0.A sequence(cn)∈ORVs is called 𝒪-regularly
varying in the Schmidt sense (i.e., it
belongs to the class IRVs),
if(1.4)lim¯λ→1k¯c(λ)=1.The classes of sequencesORVs and IRVs have an
important place in the qualitative analysis of sequential divergent processes
(see, e.g., [11–14]). Asymptotic properties of
sequences (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 senseRVs and similarly the class of slowly varying sequences
in the Karamata sense SVs are defined analogously to the classes RVf and SVf. They are fundamenatal in the theory of sequentional
regular variability in general (see [16]).Next, let(cn) be a strictly
increasing, unbounded sequence of positive numbers. Then(1.5)δc(x)=max{n∈ℕ∣cn≤x},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~nlnn, n→+∞((pn)∈IRVs) and since δp(x)~(x/lnx), 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, unbounded
sequences fromIRVs,  such that the numerical function of every one of its
elements belongs to  IRVf?The next example shows that this question has some
sense.Example 1.1.
Definec1=ln2/2 and cn=lnn, for n≥2. Then (cn) is a strictly
increasing, unbounded sequence of positive numbers. Since lnx, x≥2 belongs to the
functional class SVf (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, and h(n)=cn(n∈N) (see, e.g.,
[17]), we can assume
that h(x)=lnx for x≥2, while for x∈[1,2) we can suppose
that h is linear and
continuous on [1,2] such that h(1)=ln2/2. Therefore, δc(x)~ex, as x→+∞, so that δc belongs to de
Haan class of rapidly varying functions with index +∞ (the class R∞,f) (see, e.g.,
[19]). Hence, if λ>1 we
have(1.6)limx→+∞δc(λx)δc(x)=+∞,so that δc does not belong
to IRVf.Knowing of asymptotic characteristics of a considered
sequence and of its numerical function can be of a great importance in many
constructions of the asymptotic analysis (see, e.g., [17]).Next, we say that a functionf:[a,+∞)↦(0,+∞), a>0, belongs to the class ARVf if it is
measurable and for every λ>1 we
have(1.7)k¯f(λ)=lim¯x→+∞f(λx)f(x)>1.The function k¯f(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(1.8)f(λx)≥c(λ)·f(x).The classARVf contains (as
proper subclasses) the class of all regularly varying functions in the Karamata
sense whose index of variability is positive as well as the class of all
rapidly varying functions in de Haan sense whose index of variabilty is +∞, but it does not contain any slowly varying function
in the Kararamata sense.We also have thatARVf∩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 object of the asymptotic and the functional analysis (see,
e.g., [20]).Next, letARVs be the class of
all positive numbers (cn) such that for
every λ>1 we
have(1.9)k¯c(λ)=lim¯n→+∞c[λn]cn>1.The function k¯c(λ), λ>0, is called the auxiliary index function of the sequence (cn).The above condition is equivalent with fact that there
is ann0=n0(λ)∈N and a function c(λ)>1, λ>1, such that for every λ>1 and for every n≥n0 we
have(1.10)c[λn]≥c(λ)·cn.The classARVs contains (as
proper subclasses) the class of all regularly varying sequences in the Karamata
sense whose index of variability is positive as well as the class of all
rapidly varying sequences in de Haan sense whose index is +∞, but does not contain any slowly varying sequence in
the Karamata sense (see [21, 22]).We also have thatARVs∩IRVs≠∅ and ARVsΔIRVs≠∅.


## 2. Main Results

The next theorem is a theorem of
Galambos-Bojanic-Seneta type (see [16, 18]) for classes ARVs and ARVf. The analogous theorems for regularly varying
sequences and functions in 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 sequences and 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 as follows:
(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 lim¯n→+∞(c[λn]/cn)>1 for every λ>1. If λ>1 is arbitrary
fixed number, then k¯c(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 can
easily 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 this sequences is nonincreasing, thus Ak+1⊆Ak(k∈N) and ⋂k=1∞Ak=∅. We shall show that not all subsets Ak(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 gives that (α,α+δα)⊆Ak if α∈Ak. Assuming now that a set Ak is dense in (1,λ), we get that the 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 second
category. Then we get that the set ⋂k=1∞Ak 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 dense in (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 everyα∈[A,B] we have nα≤n0. Hence, for every n≥n0≥nα and every α∈[A,B] we have c[αn]/cn>1. Consequently, for any λ∈(1,+∞) and all
sufficiently large x≥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] and η=2λ/(A+B). Since η>1, we get lim¯x→+∞c[λx]/c[x]≥k¯c(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 Theorem2.1) a unique development of the theory of sequences from
the class ARVs and theory of
the functions from the class ARVf. Thus, Theorem 2.1 can be used to interpret all
asymptotic behaviors of functions from the class ARVf (some of them
are given in [28]) as
behavior of sequences 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.Proof.
(a) Let(cn) be a strictly
increasing unbounded sequence of positive numbers, and assume that (cn)∈ARVs. Then by Theorem 2.1,  f(x)=c[x],x≥1, belongs to ARVf. f is
nondecreasing and unbounded for x≥1. Let f←(x)=inf{y≥1∣f(y)>x},x≥c1, be the generalized inverse (see [1]) of f. It is correctly defined nondecreasing and unbounded
function 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.
Sincef← 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 the class ARVf. So by Theorem 2.1 we get that (cn)∈ARVc.
(b) Now, assume that(cn) is a strictly
increasing unbounded sequence of positive numbers and (cn)∈IRVs. Then by [13], f(x)=c[x], x≥1, belongs to IRVf. Analogously to (a), then δc(x)=f←(x)−1, x≥c1. According to [29] (or [28]) we have that function f←(x), x≥c1, belongs to the class ARVf, and consequently δc∈ARVf.
Next, let(cn) be a strictly
increasing unbounded sequence of positive numbers, and assume 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 to the class IRVf. According to [13] the sequence (cn)(cn=f(n),n∈N) belongs to IRVs.LetKc,s*,i be the class of
all strictly increasing unbounded sequences from the class IRVs∩ARVs (see [8]). This class contains (as a
proper subclass) all strictly increasing unbounded regularly varying sequences
in the Karamata sense whose index of variability is 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 this paper. It is a corollary of Corollary2.2 (and,
indirectly, of Theorem 2.1).Corollary 2.3.
The classKc,s*,i is the largest
proper subclass of the class of strictly increasing unbounded sequences from
the class IRVs, such that the numerical function of any its element
belongs to the class IRVf.Proof.
Let(cn) be a strictly
increasing unbounded sequence of positive numbers from the class IRVs∩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 the class 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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