On the convergence of positive increasing functions to infinity
We study the conditions of convergence to infinity for some classes of functions extending the well-known class of regularly varying (RV) functions, such as, e.g., $O$-regularly varying (ORV) functions or positive increasing (PI) functions.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508960910475264 |
|---|---|
| author | Buldygin, V. V. Klesov, O. I. Steinebach, J. G. Булдигін, В. В. Клесов, О. І. Штайнебах, Й. Г. |
| author_facet | Buldygin, V. V. Klesov, O. I. Steinebach, J. G. Булдигін, В. В. Клесов, О. І. Штайнебах, Й. Г. |
| author_sort | Buldygin, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:41:19Z |
| description | We study the conditions of convergence to infinity for some classes of functions extending the well-known class of regularly varying (RV) functions, such as, e.g., $O$-regularly varying (ORV) functions or positive increasing (PI) functions. |
| first_indexed | 2026-03-24T02:33:31Z |
| format | Article |
| fulltext |
UDC 517.51
V. V. Buldygin, O. I. Klesov (Nat. Techn. Univ. Ukraine “KPI”, Kyiv),
J. G. Steinebach (Math. Inst., Univ. Köln, Germany)
ON THE CONVERGENCE TO INFINITY
OF POSITIVE INCREASING FUNCTIONS
ПРО ЗБIЖНIСТЬ ДО НЕСКIНЧЕННОСТI
ДОДАТНО ЗРОСТАЮЧИХ ФУНКЦIЙ
We study conditions for the convergence to infinity of some classes of functions extending the well-known
class of regularly varying (RV) functions, such as, for example, O-regularly varying (ORV) functions or
positive increasing (PI) functions.
Дослiджено умови збiжностi до нескiнченностi деяких класiв функцiй, що розширюють вiдомий клас
регулярно змiнних функцiй, таких, як, наприклад, O-регулярно змiнних функцiй або додатно змiнних
функцiй.
1. Introduction. There is a variety of problems in mathematical as well as in stochastic
analysis, where an ordinary or a random function under investigation is assumed to
converge to infinity when its argument tends to infinity. An example of such a problem
is the question of equivalence of the solutionsX(t) and x(t), respectively, of a stochastic
differential equation (SDE) and its corresponding ordinary differential equation (ODE)
where the ODE is obtained from the SDE after excluding the stochastic differential
therein (see [15, 21, 16]). More precisely, under a certain set of assumptionsX(t) ∼ x(t)
almost surely as t→∞ if X(t)→∞ almost surely as t→∞.
This result was derived from a generalization of Karamata’s theory of regular varia-
tion (see also [12]). Indeed, Karamata’s theory provides another area, now in mathemat-
ical analysis, in which the convergence to infinity of functions plays a crucial role.
Recall that Karamata [18], in 1930, introduced the notion of regularly varying (RV)
functions and proved a number of fundamental results for this important class of func-
tions (see also [19]). Later on, these results and their further generalizations developed
into a well-established theory having a wide range of applications (cf., e.g., Bingham et
al. [6] and Seneta [24]).
Indeed, after the seminal paper [18], it is a variety of generalizations of RV functions
that has been introduced and discussed. Certainly a first one to mention is the class of O-
regularly varying (ORV) functions due to Avakumović [3], which has also been studied
in numerous papers (see, for example, [20, 14, 1, 4, 2]). In fact, it turns out that, in
many asymptotic problems, an important role is played by classes of functions, which
generalize RV functions in one way or another.
In the current paper, we continue our earlier work in [7 – 9, 11]. Here now, main
attention is paid to the class of (so-called) positive increasing (PI) functions (see Defi-
nition 2.2 below). These and some related functions have been studied by many authors
(cf., e.g., [5, 17, 25, 13, 23], to mention just a few). Along with the classes of mea-
surable ORV and PI functions, we also consider their extensions, i.e., WORV and WPI
functions, which may be measurable or nonmeasurable functions as well.
c© V. V. BULDYGIN, O. I. KLESOV, J. G. STEINEBACH, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10 1299
1300 V. V. BULDYGIN, O. I. KLESOV, J. G. STEINEBACH
The class of PI functions contains all RV functions with positive index. One of the
important properties of the latter functions is that they tend to infinity as their argument
tends to infinity (see, for example, [6, 24]). We show below that this property retains for
all functions, which are both ORV and PI functions.
The paper is organized as follows. Section 2 contains all necessary definitions con-
cerning the classes of functions considered in this paper. In Section 3 we prove that
the upper limit of a WPI function as well as the upper limits of its limit functions are
infinite. Some extra conditions, being sufficient for WPI functions to tend to infinity, are
discussed in Section 4, where we separately treat the conditions that either contain or
not the assumption of measurability. A counterexample, given in Section 5, shows that
Theorem 4.1 cannot be improved in general.
2. Definitions and some preliminary results. Let R be the set of real numbers,
R+ the set of positive reals, N the set of positive integers, and let N0 = N ∪ {0}.
Denote by F the set of all real functions f = (f(t), t ≥ 0) and let
F+ =
⋃
a>0
{f ∈ F : f(t) > 0, t ∈ [a,∞)}.
It is clear that f ∈ F+ if and only if f(t) > 0 for all sufficiently large t.
By F(∞) we denote the set of functions f ∈ F+ such that
lim sup
t→∞
f(t) =∞,
and by F∞ we denote the subset of F(∞) such that
lim
t→∞
f(t) =∞.
Throughout the paper, “measurability” means measurability in the Lebesgue sense.
For f ∈ F+, define its upper and lower limit function, i.e.,
f∗(c) = lim sup
t→∞
f(ct)
f(t)
and f∗(c) = lim inf
t→∞
f(ct)
f(t)
, c > 0,
taking values in [0,∞].
Limit functions are useful for defining and studying various classes of real functions.
Note that f∗ and f∗ are also called the index functions of f.
The following properties of limit functions follow directly from the definitions.
Lemma 2.1. Let f ∈ F+. Then
(i) for all c > 0,
0 ≤ f∗(c) ≤ f∗(c) ≤ ∞;
(ii) for all c > 0,
f∗(c) =
1
f∗(1/c),
where 1/∞ = 0 and 1/0 =∞;
(iii) for all c1 and c2 > 0, the following inequalities hold if they do not contain an
expression 0 · ∞ or∞ · 0:
f∗(c1)f∗(c2) ≤ f∗(c1c2) ≤ min{f∗(c1)f∗(c2), f∗(c2)f∗(c1)} ≤
≤ max{f∗(c1)f∗(c2), f∗(c2)f∗(c1)} ≤ f∗(c1c2) ≤ f∗(c1)f∗(c2);
(iv) f∗(1) = f∗(1) = 1.
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ON THE CONVERGENCE TO INFINITY OF POSITIVE INCREASING FUNCTIONS 1301
ORV and RV functions. We first recall the definition of ORV functions (see [3, 20],
or [1]).
Definition 2.1. A function f ∈ F+ is called a function with O-regular variation
in the wide sense (WORV) if
0 < f∗(c) ≤ f∗(c) <∞ for all c > 1. (2.1)
Correspondingly, a measurable WORV function is called a function with O-regular
variation (ORV).
The class of all WORV (ORV) functions is denoted by WORV (ORV).
We say that a measurable function f ∈ F+ varies regularly [18, 19], or that it is an
RV function, if
f∗(c) = f∗(c) = κf (c) ∈ (0,∞) for all c > 0,
that is, the limit
κf (c) = lim
t→∞
f(ct)
f(t)
exists and is positive and finite for all c > 0.
The class of all RV functions is denoted by RV . It is clear that any RV function
belongs to the class ORV .
If f ∈ RV , then
κf (c) = κ(c, ρ) = cρ, c > 0, (2.2)
for some real number ρ called the index of the function f . By RV+ we mean the class
of all RV functions with positive index. RV functions such that ρ = 0 are called slowly
varying (SV) functions.
For any RV function f , the following representation holds:
f(t) = tρ`(t), t > 0,
where (`(t), t > 0) is a slowly varying function.
WPI and PI functions. Here we recall the definitions of WPI and PI functions
(see [5, 17, 7, 9, 11]).
Definition 2.2. A function f ∈ F+ is said to be positive increasing in the wide
sense (WPI) if
f∗(c0) > 1 for some c0 > 1. (2.3)
Correspondingly, a measurable WPI function is said to be positive increasing (PI).
The class of all PI (WPI) functions is denoted by PI (WPI). Relation (2.2) implies
that RV+ ⊂ PI.
2.1. An example. There are several cases where the limit functions f∗ and f∗
determine the asymptotics of the original function f . Say, if f∗(c) = f∗(c) for all c > 0,
then f is regularly varying. If additionally (2.3) holds, then f(x)→∞ as x→∞. This
is not the case in more general situations. Below we construct a function f depending on
two parameters 0 < θ1 < 1 and θ2 > 1 such that f∗(c) = θ1 and f∗(c) = θ2 for c > 1.
Surprisingly, the asymptotics of f only depends on the specific relation between θ1
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1302 V. V. BULDYGIN, O. I. KLESOV, J. G. STEINEBACH
and θ2. In particular, lim f(x) =∞ if θ1θ2 > 1. Otherwise, f is bounded. Note that this
function f is measurable but f 6∈ PI.
Example 2.1. Let 0 < θ1 < 1 < θ2 be two given numbers. We construct a function f
such that
f∗(c) = θ1 and f∗(c) = θ2 for all c > 1.
First, choose positive sequences {Hn} and {tn} as follows:
H1 = 1, Hn+1 = θ1θ2Hn,
t1 = 1, tn+1 = n2tn, n ≥ 1.
For simplicity, let Ln = [tn, ntn) and Rn = [ntn, tn+1). Then the function f is given
as
f(t) =
1, 0 ≤ t < t2,
Hn, t ∈ Ln, n ≥ 2,
θ1Hn, t ∈ Rn, n ≥ 2.
Let c > 1. If t is sufficiently large and t ∈ Ln ∪Rn, then ct ∈ Ln ∪Rn ∪ Ln+1. Thus,
for c > 1 and sufficiently large t,
f(ct)
f(t)
=
1, if ct, t ∈ Ln, or ct, t ∈ Rn,
θ1, if ct ∈ Rn, but t ∈ Ln,
θ2, if ct ∈ Ln+1, but t ∈ Rn.
Therefore, for all c > 1,
f∗(c) = θ1, f∗(c) = θ2.
Note that
lim f(t) =∞, if θ1θ2 > 1,
lim inf f(t) = θ1, lim sup f(t) = 1, if θ1θ2 = 1,
lim f(t) = 0, if θ1θ2 < 1.
The rest of the paper is devoted to finding some conditions imposed on limit func-
tions that imply lim sup f(x) = ∞ or lim f(x) = ∞. In doing so, the basic tool is the
WPI property (2.3).
3. Upper limits of WPI functions. The following result contains a characterization
of WPI functions in terms of their lower limit functions. In particular, it shows that
f ∈ WPI if and only if f∗ ∈ F(∞).
Proposition 3.1. For f ∈ F+, the following three conditions are equivalent:
(a) f ∈ WPI;
(b) lim supc→∞ f∗(c) =∞;
(c) lim supc→∞ f∗(c) > 1.
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ON THE CONVERGENCE TO INFINITY OF POSITIVE INCREASING FUNCTIONS 1303
Proof. If f ∈ WPI, then condition (2.3) and Lemma 2.1 imply that
lim sup
c→∞
f∗(c) ≥ lim sup
m→∞
f∗(c0
m) ≥ lim sup
m→∞
(f∗(c0))m =∞.
Therefore the implication (a) =⇒ (b) is proved. On the other hand, the implications
(b) =⇒ (c) and (c) =⇒ (a) are trivial.
Lemma 2.1 (i) and Proposition 3.1 imply that the lower and upper limit functions
belong to the class F(∞) if f ∈ WPI. Below we show that WPI functions themselves
belong to the class F(∞).
Lemma 3.1. If f ∈ WPI, then f ∈ F(∞).
Proof. Let f ∈ WPI, that is, condition (2.3) holds. This implies that there are
constants c0 > 1, t0 > 0, and r > 1 such that f(t) > 0 for t ≥ t0, and
f(c0t)
f(t)
≥ r for all t ≥ t0. (3.1)
Thus
f(cm0 t)
f(t)
=
f(cm0 t)
f(cm−10 t)
· · · f(c0t)
f(t)
≥ rm
and
f(cm0 t) ≥ rmf(t) (3.2)
for all t ≥ t0 and all m ∈ N0. This implies that
lim sup
t→∞
f(t) ≥ lim sup
m→∞
f(cm0 t0) ≥ f(t0) lim
m→∞
rm =∞.
4. Conditions for the convergence to infinity of WPI and PI functions. In view
of Lemma 3.1, a natural problem is to find conditions under which WPI functions tend
to infinity. We discuss some appropriate conditions below. The following result does not
require measurability of the function f .
Proposition 4.1. Let f ∈ WPI and c0 > 1, t0 > 0, and r > 1 be the constants
from inequality (3.1). If there exists a T0 ≥ t0 such that
ε0 = inf
T0≤θ≤c0T0
f(θ) > 0, (4.1)
then f ∈ F∞.
Proof. Inequality (3.2) implies that, for all t ≥ T0,
f(t) ≥ f
(
t
c
m(t)
0
)
rm(t),
where
m(t) = max {n ∈ N0 : cn0T0 ≤ t} .
It is clear that m(t) is the integer part of logc0 (t/T0) and
t
c
m(t)
0
∈ [T0, c0T0]
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1304 V. V. BULDYGIN, O. I. KLESOV, J. G. STEINEBACH
for all t ≥ T0. Thus
f(t) ≥ ε0rm(t), t ≥ T0. (4.2)
Since limt→∞m(t) =∞ and r > 1, we have
lim
t→∞
rm(t) =∞.
Taking condition (4.1) into account, we obtain
lim inf
t→∞
f(t) ≥ ε0 lim
t→∞
rm(t) =∞.
Remark 4.1. One can substitute any of the following two conditions for (4.1):
(i) for all sufficiently large s > 0, there exists T = T (s) ≥ s, such that
inf
T≤t≤c0T
f(t) > 0, (4.3)
or
(ii) lim inft→∞ f(t) > 0.
Each of the latter two conditions implies condition (4.1) and thus is sufficient to
show that a WPI function f tends to infinity.
In turn, condition (4.3) holds if, for example, for all sufficiently large s > 0 there
exists T ≥ s such that the positive function f is continuous on the interval [T, c0T ].
Remark 4.2. Inequality (4.2) shows that any WPI function cannot grow slower than
a power function depending on the point T0 given in condition (4.1).
Proposition 4.1 yields the following result.
Corollary 4.1. Let f ∈ WPI. If there is a T > 0 such that f is continuous on the
interval [T,∞), then f ∈ F∞.
In fact, the function f in Corollary 4.1 belongs to PI. The following result treats
this case, too.
Theorem 4.1. If f ∈ ORV ∩ PI, then f ∈ F∞.
Remark 4.3. Since ORV ∩PI = ORV ∩WPI, Theorem 4.1 means that the ORV
property, like condition (4.1) in Proposition 4.1, is also an extra condition under which
a WPI function tends to infinity. Note that the ORV property requires both the WORV
condition (2.1) and measurability of the corresponding function.
Remark 4.4. Since RV+ ⊂ ORV ∩PI, Theorem 4.1 implies that any RV function
with positive index tends to infinity (see [6, 24]).
We need an auxiliary result to prove Theorem 4.1.
Lemma 4.1. Let f ∈ ORV ∩ PI. Then
lim
n→∞
f(antn)
f(tn)
=∞ (4.4)
for all sequences of positive numbers {an} and {tn} such that
lim
n→∞
an =∞ and lim
n→∞
tn =∞.
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ON THE CONVERGENCE TO INFINITY OF POSITIVE INCREASING FUNCTIONS 1305
Proof. Let {an} and {tn} be sequences of positive numbers such that limn→∞ an =
= ∞ and limn→∞ tn = ∞. Moreover let c0 > 1, t0 > 0 and r > 1 be the constants
from inequality (3.1) corresponding to the function f .
Using {an} and c0, we construct a sequence {mn} ⊂ N such that, for some n0 ∈ N,
cmn
0 ≤ an < cmn+1
0 , n ≥ n0.
Since limn→∞ an =∞, we get limn→∞mn =∞.
It is clear that, for all n ≥ 1,
f(antn)
f(tn)
=
f(antn)
f(cmn
0 tn)
mn∏
k=1
f(ck0tn)
f(ck−10 tn)
.
This together with (3.1) implies that
f(antn)
f(tn)
≥ f(antn)
f(cmn
0 tn)
rmn
for all large n.
Since f ∈ ORV , the integral representation of ORV functions (see [1]) implies that
there exist measurable bounded functions α and β such that
f(t) = Φ(t) exp
t∫
t0
β(u)
du
u
(4.5)
for all sufficiently large t, where Φ = exp ◦α. This, for sufficiently large n, implies that
f(antn)
f(cmn
0 tn)
=
Φ(antn)
Φ(cmn
0 tn)
exp
antn∫
cmn
0 tn
β(u)
du
u
≥
≥ Φ(antn)
Φ(cmn
0 tn)
exp
{
−B ln
(
an
cmn
0
)}
≥ K c−B0 ,
where
K =
lim inft→∞Φ(t)
2 lim supt→∞Φ(t)
> 0 and B = | inf
t∈[t0,∞)
β(t)| <∞.
Therefore
lim inf
n→∞
f(antn)
f(tn)
≥
(
K c−B0
)
lim inf
n→∞
rmn =∞,
whence relation (4.4) follows.
Lemma 4.1 is proved.
Proof of Theorem 4.1. Assume the converse, that is, let f(t) not tend to ∞ as
t → ∞. Then there is a sequence of positive numbers {un} and a number p ∈ [0,∞)
such that un →∞ and limn→∞ f(un) = p.
If p ∈ (0,∞), there exists a sequence of natural numbers {nk} such that, with
sk = unk
, k ≥ 1, we have limk→∞ sk =∞ and
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1306 V. V. BULDYGIN, O. I. KLESOV, J. G. STEINEBACH
lim
k→∞
sk+1/sk =∞ and lim
k→∞
f(sk+1)/f(sk) = 1.
If p = 0, there is a sequence of natural numbers {nk} such that, with sk = unk
,
k ≥ 1, we have limk→∞ sk =∞ and
lim
k→∞
sk+1/sk =∞ and lim
k→∞
f(sk+1)/f(sk) = 0.
In both cases, this is a contradiction to Lemma 4.1, which completes the proof of
f ∈ F∞.
5. Counterexamples. In this section, we discuss two examples highlighting the
sufficiency conditions of our preceding results.
5.1. Measurability. Analyzing the proof of Theorem 4.1, we see that the WORV
condition (2.1) can be weakened. Indeed, Theorem 4.1 remains true for measurable
functions f for which the integral representation (4.5) holds for sufficiently large t,
where the measurable function α is bounded and the measurable function β is bounded
from below and locally bounded from above. This rises the interesting question whether
or not one can drop the measurability in Theorem 4.1 and instead add the following
“one-sided” WORV condition to the WPI condition (2.3):
f∗(c) > 0 for all c > 1,
or, more stronger,
f∗(c) > 1 for all c > 1
(implying the WPI condition (2.3)).
The following result shows that this is not the case.
Proposition 5.1. There exists a nonmeasurable function f ∈ F+ such that
lim
t→∞
f(ct)
f(t)
=∞ for all c > 1, (5.1)
but
lim inf
t→∞
f(t) = 0. (5.2)
Proof. Let H be the Hamel basis (see, for example, [16, 6]), that is, a set of real
numbers such that every real number x 6= 0 can uniquely be represented as a finite linear
combination of elements of H with rational coefficients, i.e.,
x =
n(x)∑
i=1
ri(x)bi(x),
where n(x) ∈ N, ri(x) ∈ Q \ {0} and bi(x) ∈ H.
Note that (n(x), x ∈ R) is a nonmeasurable and subadditive function, that is
n(x+ y) ≤ n(x) + n(y) for all x, y ∈ R, (5.3)
(see, for example, [6] or [22]). Moreover, for all fixed n ≥ 1 and all fixed, but different
b1, . . . , bn ∈ H,
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ON THE CONVERGENCE TO INFINITY OF POSITIVE INCREASING FUNCTIONS 1307
the set Mn =
{
n∑
i=1
ribi; r1, . . . , rn ∈ Q \ {0}
}
is dense in R. (5.4)
Let
h(x) = x2 − n(x), x > 0, and f(t) = exp{h(ln t)}, t > 0.
It is clear that f ∈ F+. Moreover, inequality (5.3) implies that
h(x+ u)− h(x) = 2xu+ u2 − (n(x+ u)− n(x)) ≥ 2xu− n(u)
for all x > 0 and u > 0. Hence
limx→∞
(
h(x+ u)− h(x)
)
=∞ for all u > 0.
This implies that
lim
t→∞
f(ct)
f(t)
= lim
t→∞
exp{h(ln t+ ln c)− h(ln t)} =
= exp
{
lim
t→∞
(h(ln t+ ln c)− h(ln t))
}
=∞
for all c > 1. This proves relation (5.1).
On the other hand, according to (5.4), there exists a sequence {xk} such that
xk ∈ (k − 1, k) ∩Mk3 , k ≥ 1.
It is clear that
h(xk) < k2 − n(xk) = k2 − k3 for all k ≥ 1,
whence
lim inf
x→∞
h(x) = −∞.
This implies (5.2) and thus completes the proof of Proposition 5.1.
5.2. Upper limit functions. The sufficient condition of Theorem 4.1 for f ∈ F∞
is expressed in terms of the lower limit function f∗. Note that the upper limit function,
in turn, is not an appropriate tool here. The following example exhibits a bounded
measurable function f such that f∗(c) = ∞ for all c > 0. One may compare this
function with g(x) = ex for which the upper limit function is nearly the same, i.e.,
g∗(c) =∞ for c > 1, but g grows to infinity very fast.
Example 5.1. Let B = {1!, 2!, 3!, . . . } and A = R+ \B. Put
f(t) = 1IA(t) +
∞∑
n=1
1
n
1I{n!}(t).
In other words,
f(t) =
1, t ∈ A,
1
n
, t = n! for some n.
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1308 V. V. BULDYGIN, O. I. KLESOV, J. G. STEINEBACH
For any c > 0, the case where both ct = n! and t = m! does not occur if t is sufficiently
large. Then, for c > 0 and sufficiently large t,
f(ct)
f(t)
=
1, ct ∈ A, t ∈ A,
1
n
, ct = n! for some n, but t ∈ A,
n, ct ∈ A, but t = n! for some n.
Therefore f∗(c) = 0 and f∗(c) =∞ for all c > 0. However 0 < f(t) ≤ 1.
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Received 14.10.09,
after revision — 30.07.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
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| id | umjimathkievua-article-2956 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:33:31Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/11/7d3125d688b0119ef5cf55d805473811.pdf |
| spelling | umjimathkievua-article-29562020-03-18T19:41:19Z On the convergence of positive increasing functions to infinity Про збіжність до нескінченності додатно зростаючих функцій Buldygin, V. V. Klesov, O. I. Steinebach, J. G. Булдигін, В. В. Клесов, О. І. Штайнебах, Й. Г. We study the conditions of convergence to infinity for some classes of functions extending the well-known class of regularly varying (RV) functions, such as, e.g., $O$-regularly varying (ORV) functions or positive increasing (PI) functions. Досліджено умови збіжності до нескінченності деяких класів функцій, що розширюють відомий клас регулярно змінних функцій, таких, як, наприклад, $O$-регулярно змінних функцій або додатно змінних функцій. Institute of Mathematics, NAS of Ukraine 2010-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2956 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 10 (2010); 1299–1308 Український математичний журнал; Том 62 № 10 (2010); 1299–1308 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2956/2662 https://umj.imath.kiev.ua/index.php/umj/article/view/2956/2663 Copyright (c) 2010 Buldygin V. V.; Klesov O. I.; Steinebach J. G. |
| spellingShingle | Buldygin, V. V. Klesov, O. I. Steinebach, J. G. Булдигін, В. В. Клесов, О. І. Штайнебах, Й. Г. On the convergence of positive increasing functions to infinity |
| title | On the convergence of positive increasing functions to infinity |
| title_alt | Про збіжність до нескінченності додатно зростаючих функцій |
| title_full | On the convergence of positive increasing functions to infinity |
| title_fullStr | On the convergence of positive increasing functions to infinity |
| title_full_unstemmed | On the convergence of positive increasing functions to infinity |
| title_short | On the convergence of positive increasing functions to infinity |
| title_sort | on the convergence of positive increasing functions to infinity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2956 |
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