Abstract

In the theorems of Galambos-Bojanić-Seneta's type, the asymptotic behavior of the functions , for , is investigated by the asymptotic behavior of the given sequence of positive numbers , as and vice versa. The main result of this paper is one theorem of such a type for sequences of positive numbers which satisfy an asymptotic condition of the Karamata type , for .

1. Introduction

A function is called -regularly varying in the sense of Karamata (see [1]) if it is measurable and if for every ,

Function is called the index function of , and is the class of all -regularly varying functions defined on some interval .

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

-regularly varying functions in the Schmidt sense form the functional class and (see [3]). They represent an important object in the analysis of divergent processes (see [49]). In particular, we have that the class of regularly varying functions in the Karamata sense satisfies (see [3], and some of its applications can be found in [10]).

A function is called regularly varying in Karamata sense if for every and a fixed . If , then is called slowly varying in the Karamata sense , and all such functions form the class . We have that (see [10]).

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

A sequence is called -regularly varying in the Schmidt sense (i.e., it belongs to the class ), if

The classes of sequences and have an important place in the qualitative analysis of sequential divergent processes (see, e.g., [1114]). 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 and similarly the class of slowly varying sequences in the Karamata sense are defined analogously to the classes and . They are fundamenatal in the theory of sequentional regular variability in general (see [16]).

Next, let be a strictly increasing, unbounded sequence of positive numbers. Thenfor , is the numerical function of the sequence (see, e.g., [17]).

In the sequel, let be the strong asymptotic equivalence of sequences and functions, and let be the sequence of prime numbers in the increasing order. Since , and since , , () (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 , such that the numerical function of every one of its elements belongs to ?

The next example shows that this question has some sense.Example 1.1. Define and , for . Then is a strictly increasing, unbounded sequence of positive numbers. Since , belongs to the functional class (see, e.g., [10]), by a result from [18], we have that . Hence, . Next, since , , where , , is continuous and strictly increasing, and (see, e.g., [17]), we can assume that for , while for we can suppose that is linear and continuous on such that . Therefore, , as , so that belongs to de Haan class of rapidly varying functions with index (the class ) (see, e.g., [19]). Hence, if we haveso that does not belong to .

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 , , belongs to the class if it is measurable and for every we haveThe function , , is the auxiliary index function of the function , .

Condition (1.7) is equivalent with assumption that there exists an and for , so that for every and every it holds

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

Next, let be the class of all positive numbers such that for every we haveThe function , , is called the auxiliary index function of the sequence .

The above condition is equivalent with fact that there is an and a function , , such that for every and for every we have

The class 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 and .

2. Main Results

The next theorem is a theorem of Galambos-Bojanic-Seneta type (see [16, 18]) for classes and . The analogous theorems for regularly varying sequences and functions in the Karamata sense, -regularly varying sequences and functions in the Karamata sense, sequences from the class and functions from the class , 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, 2427].Theorem 2.1. Let be a sequence of positive numbers. Then the next assertions are equivalent as follows:
(a),(b), , belongs to the class .
Proof. Let be a sequence of positive numbers and assume that , thus that for every . If is arbitrary fixed number, then for every . For arbitrary define in the following way: if for every , and else. One can easily see that for every considered .
Next, define a sequence of sets by . Then this sequences is nonincreasing, thus and . We shall show that not all subsets are dense in . If for a fixed , then , and there is a such that for every . Hence, every belongs to , since . This gives that if . Assuming now that a set is dense in , we get that the set is also dense in . If else, we assume that all sets are dense in , we find that is a sequence of open dense subsets of the set of the second category. Then we get that the set is dense in , so it must be nonempty, which is a contradiction. Hence, we conclude that there is an , so that the set is not dense in . Hence, there is an intervals such that .
Therefore, for every we have . Hence, for every and every we have . Consequently, for any and all sufficiently large we have that where and . Since , we get , so that belongs to the class .
Since 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 and theory of the functions from the class . Thus, Theorem 2.1 can be used to interpret all asymptotic behaviors of functions from the class (some of them are given in [28]) as behavior of sequences from the class and vice versa.Corollary 2.2. Let be a strictly increasing unbounded sequence of positive numbers. Then,
(a) if and only if ;(b) if and only if .
Proof. (a) Let be a strictly increasing unbounded sequence of positive numbers, and assume that . Then by Theorem 2.1, , belongs to . is nondecreasing and unbounded for . Let , be the generalized inverse (see [1]) of . It is correctly defined nondecreasing and unbounded function for . It is also stepwise and right continuous. We also have that for .
According to [22] we have that function , , belongs to the class .
Since is nondecreasing and unbounded, we get , so that , , belongs to .
Next, let be a strictly increasing unbounded sequence of positive numbers, and let , , belong to . Besides, let , . Since for , we find that . According to [28] we have that function , , belongs to the class . So by Theorem 2.1 we get that .
(b) Now, assume that is a strictly increasing unbounded sequence of positive numbers and . Then by [13], , , belongs to . Analogously to (a), then , . According to [29] (or [28]) we have that function , , belongs to the class , and consequently .
Next, let be a strictly increasing unbounded sequence of positive numbers, and assume that . Besides, let , . Since , for , then . According to [29] (or [28]) we have that function , , belongs to the class . According to [13] the sequence belongs to .

Let be the class of all strictly increasing unbounded sequences from the class (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 , nor from the class .

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 is the largest proper subclass of the class of strictly increasing unbounded sequences from the class , such that the numerical function of any its element belongs to the class .Proof. Let be a strictly increasing unbounded sequence of positive numbers from the class . Then by Corollaries 2.2(a) and 2.2(b), , thus . Next, assume that is a strictly increasing unbounded sequence of positive numbers from the class . Then by Corollary 2.2(b), , and by Corollary 2.2(a) .
This completes the proof.