A Theorem of Galambos-Bojanić-Seneta Type

Abstract

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

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, 𝑘_𝑓(𝜆)∶=lim_𝑥→+∞𝑓(𝜆𝑥)𝑓(𝑥)<+∞. (1.1)

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

A function 𝑓∈ORV_𝑓 is called 𝒪-regularly varying in the Schmidt sense (see [2, 3]) iflim_𝜆→1𝑘_𝑓(𝜆)=1. (1.2)

𝒪-regularly varying functions in the Schmidt sense form the functional class IRV_𝑓 and IRV_𝑓⊊ORV_𝑓 (see [3]). They represent an important object in the analysis of divergent processes (see [4–9]). In particular, we have that the class RV_𝑓 of regularly varying functions in the Karamata sense satisfies RV_𝑓⊊IRV_𝑓 (see [3], and some of its applications can be found in [10]).

A function 𝑓∈IRV_𝑓 is called regularly varying in Karamata sense if 𝑘_𝑓(𝜆)=𝜆^𝜌 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 SV_𝑓. We have that SV_𝑓⊊RV_𝑓 (see [10]).

A sequence of positive numbers (𝑐_𝑛) is called 𝒪-regularly varying in the Karamata sense (i.e., it belongs to the class ORV_𝑠), if𝑘_𝑐(𝜆)=lim_𝑛→+∞𝑐_[𝜆𝑛]𝑐_𝑛<+∞, (1.3)for every 𝜆>0.

A sequence (𝑐_𝑛)∈ORV_𝑠 is called 𝒪-regularly varying in the Schmidt sense (i.e., it belongs to the class IRV_𝑠), iflim_𝜆→1𝑘_𝑐(𝜆)=1. (1.4)

The classes of sequences ORV_𝑠 and IRV_𝑠 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 sense RV_𝑠 and similarly the class of slowly varying sequences in the Karamata sense SV_𝑠 are defined analogously to the classes RV_𝑓 and SV_𝑓. They are fundamenatal in the theory of sequentional regular variability in general (see [16]).

Next, let (𝑐_𝑛) be a strictly increasing, unbounded sequence of positive numbers. Then𝛿_𝑐(𝑥)=max{𝑛∈ℕ∣𝑐_𝑛≤𝑥}, (1.5)for 𝑥≥𝑐₁, is the numerical function of the sequence (𝑐_𝑛) (see, e.g., [17]).

In the sequel, let ∼ be the strong asymptotic equivalence of sequences and functions, and let (𝑝_𝑛) be the sequence of prime numbers in the increasing order. Since 𝑝_𝑛∼𝑛ln𝑛, 𝑛→+∞((𝑝_𝑛)∈IRV_𝑠) and since 𝛿_𝑝(𝑥)∼(𝑥/ln𝑥), 𝑥→+∞, (𝛿_𝑝∈IRV_𝑓) (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 from IRV_𝑠, such that the numerical function of every one of its elements belongs to IRV_𝑓?

The next example shows that this question has some sense.Example 1.1. Define 𝑐₁=ln2/2 and 𝑐_𝑛=ln𝑛, for 𝑛≥2. Then (𝑐_𝑛) is a strictly increasing, unbounded sequence of positive numbers. Since ln𝑥, 𝑥≥2 belongs to the functional class SV_𝑓 (see, e.g., [10]), by a result from [18], we have that (𝑐_𝑛)∈SV_𝑠. Hence, (𝑐_𝑛)∈IRV_𝑠. Next, since 𝛿_𝑐(𝑥)∼ℎ⁻¹(𝑥), 𝑥→+∞, where ℎ(𝑥), 𝑥≥1, is continuous and strictly increasing, and ℎ(𝑛)=𝑐_𝑛(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, 𝛿_𝑐(𝑥)∼e^𝑥, as 𝑥→+∞, so that 𝛿_𝑐 belongs to de Haan class of rapidly varying functions with index +∞ (the class 𝑅_∞,𝑓) (see, e.g., [19]). Hence, if 𝜆>1 we havelim_𝑥→+∞𝛿_𝑐(𝜆𝑥)𝛿_𝑐(𝑥)=+∞, (1.6)so that 𝛿_𝑐 does not belong to IRV_𝑓.

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 function 𝑓∶[𝑎,+∞)↦(0,+∞), 𝑎>0, belongs to the class ARV_𝑓 if it is measurable and for every 𝜆>1 we have𝑘_𝑓(𝜆)=lim_𝑥→+∞𝑓(𝜆𝑥)𝑓(𝑥)>1. (1.7)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 holds𝑓(𝜆𝑥)≥𝑐(𝜆)⋅𝑓(𝑥). (1.8)

The class ARV_𝑓 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 that ARV_𝑓∩IRV_𝑓≠∅ and ARV_𝑓ΔIRV_𝑓≠∅. Besides, the class ARV_𝑓 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, let ARV_𝑠 be the class of all positive numbers (𝑐_𝑛) such that for every 𝜆>1 we have𝑘_𝑐(𝜆)=lim_𝑛→+∞𝑐_[𝜆𝑛]𝑐_𝑛>1. (1.9)The function 𝑘_𝑐(𝜆), 𝜆>0, is called the auxiliary index function of the sequence (𝑐_𝑛).

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 have𝑐_[𝜆𝑛]≥𝑐(𝜆)⋅𝑐_𝑛. (1.10)

The class ARV_𝑠 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 that ARV_𝑠∩IRV_𝑠≠∅ and ARV_𝑠ΔIRV_𝑠≠∅.

2. Main Results

The next theorem is a theorem of Galambos-Bojanic-Seneta type (see [16, 18]) for classes ARV_𝑠 and ARV_𝑓. 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 IRV_𝑠 and functions from the class IRV_𝑓, 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 (𝑐_𝑛) be a sequence of positive numbers. Then the next assertions are equivalent as follows:
(a)(𝑐_𝑛)∈ARV_𝑠,(b)𝑓(𝑥)=𝑐_[𝑥], 𝑥≥1, belongs to the class ARV_𝑓.Proof. (a)⇒(b) Let (𝑐_𝑛) be a sequence of positive numbers and assume that (𝑐_𝑛)∈ARV_𝑠, thus that lim_𝑛→+∞(𝑐_[𝜆𝑛]/𝑐_𝑛)>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 𝑛_𝛼∈𝑁 in the following way: 𝑛_𝛼=1 if 𝑐_[𝛼𝑛]/𝑐_𝑛>1 for every 𝑛∈𝑁, and 𝑛_𝛼=1+max{𝑛∈𝑁∣𝑐_[𝛼𝑛]/𝑐_𝑛≤1} else. One can easily see that 1≤𝑛_𝛼<+∞ for every considered 𝛼.
Next, define a sequence of sets (𝐴_𝑘) by 𝐴_𝑘={𝛼∈(1,𝜆)∣𝑛_𝛼>𝑘}(𝑘∈𝑁). Then this sequences is nonincreasing, thus 𝐴_𝑘+1⊆𝐴_𝑘(𝑘∈𝑁) and ⋂^∞_𝑘=1𝐴_𝑘=∅. We shall show that not all subsets 𝐴_𝑘(𝑘∈𝑁) are dense in (1,𝜆). If 𝛼∈𝐴_𝑘 for a fixed 𝑘∈𝑁, then 𝑐_{[(𝑛𝛼−1)𝛼]}/𝑐_𝑛𝛼−1≤1, and there is a 𝛿_𝛼>0 such that 𝑐_{[(𝑛𝛼−1)𝑡]}/𝑐_𝑛𝛼−1=𝑐_{[(𝑛𝛼−1)𝛼]}/𝑐_𝑛𝛼−1≤1, for every 𝑡∈[𝛼,𝛼+𝛿_𝛼)⊊(1,𝜆). Hence, every 𝑡∈(𝛼,𝛼+𝛿_𝛼) belongs to 𝐴_𝑘, since 𝑛_𝑡≥(𝑛_𝛼−1)+1>𝑘. This gives that (𝛼,𝛼+𝛿_𝛼)⊆𝐴_𝑘 if 𝛼∈𝐴_𝑘. Assuming now that a set 𝐴_𝑘 is dense in (1,𝜆), we get that the set Int𝐴_𝑘 is also dense in (1,𝜆). If else, we assume that all sets 𝐴_𝑘(𝑘∈𝑁) are dense in (1,𝜆), we find that (Int𝐴_𝑘) is a sequence of open dense subsets of the set (1,𝜆) of the second category. Then we get that the set ⋂^∞_𝑘=1𝐴_𝑘 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 𝐴_𝑛0 is not dense in (1,𝜆). Hence, there is an intervals [𝐴,𝐵]⊊(1,𝜆)(𝐴<𝐵) such that [𝐴,𝐵]⊆(1,𝜆)⧵𝐴_𝑛0={𝛼∈(1,𝜆)∣𝑛_𝛼≤𝑛₀}.
Therefore, for every 𝛼∈[𝐴,𝐵] we have 𝑛_𝛼≤𝑛₀. Hence, for every 𝑛≥𝑛₀≥𝑛_𝛼 and every 𝛼∈[𝐴,𝐵] we have 𝑐_[𝛼𝑛]/𝑐_𝑛>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 ARV_𝑓.
Since (b)⇒(a) is immediate, we completed the proof.

The above theorem provides (analogously, as in cases given before Theorem 2.1) a unique development of the theory of sequences from the class ARV_𝑠 and theory of the functions from the class ARV_𝑓. Thus, Theorem 2.1 can be used to interpret all asymptotic behaviors of functions from the class ARV_𝑓 (some of them are given in [28]) as behavior of sequences from the class ARV_𝑠 and vice versa.Corollary 2.2. Let (𝑐_𝑛) be a strictly increasing unbounded sequence of positive numbers. Then,
(a)(𝑐_𝑛)∈ARV_𝑠 if and only if 𝛿_𝑐(𝑥)(𝑥≥𝑐₁)∈IRV_𝑓;(b)(𝑐_𝑛)∈IRV_𝑠 if and only if 𝛿_𝑐(𝑥)(𝑥≥𝑐₁)∈ARV_𝑓.Proof. (a) Let (𝑐_𝑛) be a strictly increasing unbounded sequence of positive numbers, and assume that (𝑐_𝑛)∈ARV_𝑠. Then by Theorem 2.1, 𝑓(𝑥)=𝑐_[𝑥],𝑥≥1, belongs to ARV_𝑓. 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 𝛿_𝑐(𝑥)=𝑓^←(𝑥)−1 for 𝑥≥𝑐₁.
According to [22] we have that function 𝑓^←(𝑥), 𝑥≥𝑐₁, belongs to the class IRV_𝑓.
Since 𝑓^← is nondecreasing and unbounded, we get lim_𝑥→+∞(𝛿_𝑐(𝑥)/𝑓^←(𝑥))=1, so that 𝛿_𝑐(𝑥), 𝑥≥𝑐₁, belongs to IRV_𝑓.
Next, let (𝑐_𝑛) be a strictly increasing unbounded sequence of positive numbers, and let 𝛿_𝑐(𝑥), 𝑥≥𝑐₁, belong to IRV_𝑓. Besides, let 𝑓(𝑥)=𝑐_[𝑥], 𝑥≥1. Since 𝑓^←(𝑥)=𝛿_𝑐(𝑥)+1 for 𝑥≥𝑐₁, we find that 𝑓^←∈IRV_𝑓. According to [28] we have that function 𝑓(𝑥), 𝑥≥1, belongs to the class ARV_𝑓. So by Theorem 2.1 we get that (𝑐_𝑛)∈ARV_𝑐.
(b) Now, assume that (𝑐_𝑛) is a strictly increasing unbounded sequence of positive numbers and (𝑐_𝑛)∈IRV_𝑠. Then by [13], 𝑓(𝑥)=𝑐_[𝑥], 𝑥≥1, belongs to IRV_𝑓. Analogously to (a), then 𝛿_𝑐(𝑥)=𝑓^←(𝑥)−1, 𝑥≥𝑐₁. According to [29] (or [28]) we have that function 𝑓^←(𝑥), 𝑥≥𝑐₁, belongs to the class ARV_𝑓, and consequently 𝛿_𝑐∈ARV_𝑓.
Next, let (𝑐_𝑛) be a strictly increasing unbounded sequence of positive numbers, and assume that 𝛿_𝑐∈ARV_𝑓. Besides, let 𝑓(𝑥)=𝑐_[𝑥], 𝑥≥1. Since 𝑓^←(𝑥)=𝛿_𝑐(𝑥)+1, for 𝑥≥𝑐₁, then 𝑓^←∈ARV_𝑓. According to [29] (or [28]) we have that function 𝑓(𝑥), 𝑥≥1, belongs to the class IRV_𝑓. According to [13] the sequence (𝑐_𝑛)(𝑐_𝑛=𝑓(𝑛),𝑛∈𝑁) belongs to IRV_𝑠.

Let 𝐾^∗,𝑖_𝑐,𝑠 be the class of all strictly increasing unbounded sequences from the class IRV_𝑠∩ARV_𝑠 (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 SV_𝑠, 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 Corollary 2.2 (and, indirectly, of Theorem 2.1).Corollary 2.3. The class 𝐾^∗,𝑖_𝑐,𝑠 is the largest proper subclass of the class of strictly increasing unbounded sequences from the class IRV_𝑠, such that the numerical function of any its element belongs to the class IRV_𝑓.Proof. Let (𝑐_𝑛) be a strictly increasing unbounded sequence of positive numbers from the class IRV_𝑠∩ARV_𝑠. Then by Corollaries 2.2(a) and 2.2(b), 𝛿_𝑐∈IRV_𝑓∩ARV_𝑓, thus 𝛿_𝑐∈IRV_𝑓. Next, assume that (𝑐_𝑛) is a strictly increasing unbounded sequence of positive numbers from the class IRV_𝑠⧵ARV_𝑠. Then by Corollary 2.2(b), 𝛿_𝑐∈ARV_𝑓, and by Corollary 2.2(a) 𝛿_𝑐∉IRV_𝑓.
This completes the proof.