Mathematica Moravica
Vol. 12 -1 (2008), 15–18
Note on Sequence of Exponents
of SO-Regular Variability
Dragan Ðurčić and Mališa Žižović
Dedicated to Professor Dušan Adamović (1928-2008)
Abstract. In this paper we develop the concept of the sequence of exponents of SO-regular variability [9], as a generalization of the sequence
of convergence exponents [1].
1. Introduction
Let (an ) be a nondecreasing sequence of positive numbers. If a(t) =
P+∞
n
n=1 an t , then it is well known (see [5]) that the asymptotic property
(1)
a(t)
< +∞
t→1− a(t2 )
lim
is equivalent with property
a[λn]
(2)
lim
= ka (λ) < +∞,
n→+∞ an
λ > 0.
Asymptotic condition (2), in the set of sequences of positive numbers,
defines the class of O-regularly varying sequences, i.e. the sequential class
OR (see [4]). Karamata’s theory of O-regular variability is an essential part
of analysis of divergence (see [2]).
An O-regularly varying sequence (an ) is called SO-regularly varying (see
[9, 5]) if there is a β ≥ 1 such that ka (λ) ≤ β, for all λ > 0. The class
of SO-regularly varying sequences is denoted by with SO, and the class of
convergence sequences of positive numbers with non zero limit is denoted by
K. For classes K, SO and OR the next relations hold:
(3)
K ⊆ SO ⊆ OR,
K 6= SO,
SO 6= OR.
In [7] Pólya and Szegö considered the concept of sequence of exponents
of convergence.
2000 Mathematics Subject Classification. Primary: 26A12.
Key words and phrases. Regular variability, Sequence of exponents of convergence.
15
c
2008
Mathematica Moravica
16
Note on Sequence of Exponents of SO-Regular Variability
Definition 1. If (cn ) is a sequence of positive numbers converging to zero,
then a sequence of positive numbers (λn ) is a sequence of exponents of
P
λn (1+ε)
convergence for the sequence (an ) if for every ε > 0 the series +∞
n=1 an
P+∞ λn (1−ε)
converges, and the series n=1 an
diverges.
The concept defined above partialy appeared in the papers [3] and [8].
Basic properties of this notion are given in [1] and [10]. Using the idea of
definition 1 we now define notion of the sequence of exponents of SO-regular
varying convergence.
Definition 2. If (cn ) is a sequence of positive numbers, then a sequence of
real numbers (λn ) is sequence of exponents of SO-regular variability, if for
P
λ (1+ε)
, n ∈ N, belong to the class SO,
all ε ≥ 0, sequence (s1n ), s1n = nk=1 ck k
P
λ (1+µ)
n
for n ∈ N, is not in the
and for all µ < 0, sequence (s2n ), s2n = k=1 cnn
class SO.
It is clear that for every sequence of positive numbers (an ) (an 6= 1,
n ≥ n0 , n0 ∈ N) there exists some sequence of exponents of SO-regularly
variability. Also, many properties of sequences of exponents of SO-regular
variability can be derived from the corresponding properties of sequences of
exponents of convergence (see [1]).
Definition 3. Sequence (an ) is potentialy O-regular varying (then we say
that (an ) belongs to the class P O), if there exists a real number ρ and a
sequence (sn ) ∈ SO such that an = nρ · sn , n ∈ N. P Oρ is the set of all
sequences which belong to the class P O for some fixed number ρ.
2. Results
Lemma 1. Let (an ) be a sequence of positive numbers and let bn = aλnn ,
n ∈ N, belongs to the class P O−1 . Then (λn ) is a sequence of exponents of
SO-regular variability for the sequence (an ).
Proof. The sequence (dn ), dn = nbn , n ∈ N, belongs to the class SO. For
d[λn]
any δ > 1 limn→+∞ supλ∈[1,δ]
= M (δ) < +∞, M (δ) ≥ 1 holds. Let
dn
f (x) = d[x] , x ≥ 1, be function generated by sequence (dn ). It is clear that
relation
f (λx)
lim
≤ kd (λ) · M (δ),
x→+∞ f (x)
for any λ > 0, δ > 1, holds. Let be M = limδ→1− M (δ). So, there exists
f (λx)
γ = β · M ≥ 1 such that limx→+∞
≤ γ for all λ > 0. Also, f (x) =
f (x)
l(x) · B0 (x), x ≥ 1, where l(x) is a slow varying function, and there exists
1
A > 0 such that
≤ B0 (x) ≤ A, x ≥ 1. If g(t) = b[t] , t ≥ 1, then we have
A
17
Dragan Ðurčić and Mališa Žižović
g(t) = [t]−1 l(t)B0 (t), t ≥ 1, and also, for any n ∈ N,
Z n+1
n
X
[t]−1 · l(t)dt = B1 (n) · l1 (n) (n ∈ N ),
bk = B1 (n) ·
1
k=1
1
where l1 (t), t ≥ 1, is a slow varying function and
≤ B1 (t) ≤ A, t ≥ 1. So
A
Pn
the sequence ( k=1 bk ) is an element of the class SO.
If µ < 0, then for n ≥ 1 we have
Z n+1
n
X
l1+µ (t)
1+µ
bn = B2 (n)
[t]−1
dt = B2 (n)l2 (n)n−µ ,
µ
[t]
1
k=1
where l2 (t), t ≥ 1, is slow varying function and
min{A1+µ , A−1−µ } ≤ B2 (t) ≤ max{A1+µ , A−1−µ },
t ≥ 1.
(s2n )
(def. 2) is not element of class SO.
So, sequence
If ε > 0, then for n ≥ 1 we have
Z n+1
n
X
ε l1+ε (t)
1+ε
bk = B3 (n) ·
[t]−1− 2
dt = B3 (n) · p1 (n),
ε/2
[t]
1
k=1
where p1 (t), t ≥ 1, is a function which converges to a positive number for
1
t → +∞, and 1+ε ≤ B3 (t) ≤ A1+ε , t ≥ 1. So, sequence (s1n ) belongs to
A
the class SO.
Theorem 1. Let (an ) be a sequence of positive numbers and let bn = aλnn ,
n ∈ N, belong to the class PO. The sequence (λn ) is a sequence of exponents
of SO-regular variability of sequence (an ) if and only if (bn ) belongs to the
class P O−1 , i.e.,
)
(
n
X
δ
k
bn = n−1 exp αn +
, n ≥ 1,
k
k=1
where sequence (αn ) is bounded, and sequence (δn ) converging to zero.
Proof. If (bn ) is an element of the class P O−1 , then by Lemma 1, the sequence (λn ) is a sequence of exponents of SO-regular variability for sequence
(an ). If sequence (bn ) is an element of class P Oρ , ρ > −1, then bn = nρ · sn ,
n ∈ N, where (sn ) belongs to class SO. In this case for n ≥ 1 we have
Z n+1
n
X
bn = B4 (n)
[t]ρ l(t)dt = B4 (n) · nρ+1 l3 (n),
k=1
1
1
≤ B4 (t) ≤ A, t ≥ 1. So,
A
the sequence (λn ) is not a sequence of exponents
P of SO-regular variability
for the sequence (an ), because the sequence ( nk=1 bk ) is not an element of
the class SO.
where l3 (t), t ≥ 1, is a slow varying function, and
Note on Sequence of Exponents of SO-Regular Variability
18
If (bn ) belongs to the class P Oρ , ρ < −1, then bn = nρ Sn , n ∈ N, where
p
(sn ) is an element of class SO. Then, p = −1 − ρ > 0 and for µ =
<0
2ρ
we have
n+1
Z n+1
Z
n
X
p
1+µ
ρ(1+µ) 1+µ
bk = B5 (n)
[t]
l
(t)dt = B5 (n) ·
[t]ρ+ 2 l1+µ (t)dt =
1
k=1
1
Z
= B5 (n) ·
1
n+1
3p l1+µ (t)
[t]ρ+ 4
p dt
[t] 4
= B5 (n) · p2 (n),
where p2 (t), t ≥ 1, is a function which converges to a positive limit as t
1
converges to +∞, and 1+µ ≤ B5 (t) ≤ A1+µ , t ≥ 1. So, sequence (λn ) is
A
not a sequence of exponents of SO-regular variability for (an ), because for
p
µ=
< 0, by the above assumptions, the sequence (s2n ) belongs to the
2ρ
class SO. If (bn ) is an element of class P O−1 , then bn = n−1 · l(n) · B0 (n),
n ≥ 1, where l and B
0 are functionsfrom the proof of Lemma 1. So, for
P
δk
−1
n ≥ 1, bn = n · exp αn + nk=1
where the sequence (αn ) is bounded
k
and the sequence (δn ) converges to zero.
References
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Vesnik, 5 (20) (1968), 447-458.
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Press, Cambridge, 1987.
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149-167.
[7] G. Pólya, Szegö, Aufgaben und Lehrshtze, Bd. I, 1925, 19-20.
[8] A. Pringsheim, Elementare Theorie der ganzen transcedenten Functionen von
endlicher Ordnung, Math. Ann. 58 (1904), 257-342.
[9] E. Seneta, D. Drasin, A generalization of slowly varying functions, Proc. Am. Math.
Soc. 96 (3) 1986, 470-472.
[10] M.R. Tasković, On parameter convergence and convergence in larger sense, Mat.
Vesnik, 8(23) (1971), 55-59.
Dragan Ðurčić, Mališa Žižović
Technical Faculty
Svetog Save 65
32000 Čačak
Serbia
E-mail address: [email protected]
E-mail address: [email protected]