On Cèsaro and Copson norms of nonnegative sequences
The C`esaro and Copson norms of a nonnegative sequence are lp-norms of its arithmetic means and the corresponding conjugate means. It is well known that, for $1 < p < \infty$, these norms are equivalent. In 1996, G. Bennett posed the problem of finding the best constants in the associa...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507184275652608 |
|---|---|
| author | Kolyada, V. I. Коляда, В. І. |
| author_facet | Kolyada, V. I. Коляда, В. І. |
| author_sort | Kolyada, V. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:54:43Z |
| description | The C`esaro and Copson norms of a nonnegative sequence are lp-norms of its arithmetic means and the corresponding
conjugate means. It is well known that, for $1 < p < \infty$, these norms are equivalent. In 1996, G. Bennett posed the problem
of finding the best constants in the associated inequalities. The solution of this problem requires the evaluation of four
constants. Two of them were found by G. Bennett. We find one of the two unknown constants and also prove one optimal
weighted-type estimate regarding the remaining constant. |
| first_indexed | 2026-03-24T02:05:17Z |
| format | Article |
| fulltext |
UDC 517.5
V. I. Kolyada (Karlstad Univ., Sweden)
ON CÈSARO AND COPSON NORMS OF NONNEGATIVE SEQUENCES
ПРО НОРМИ ЧЕЗАРО I КОПСОНА НЕВIД’ЄМНИХ ПОСЛIДОВНОСТЕЙ
The Cèsaro and Copson norms of a nonnegative sequence are lp -norms of its arithmetic means and the corresponding
conjugate means. It is well known that, for 1 < p < \infty , these norms are equivalent. In 1996, G. Bennett posed the problem
of finding the best constants in the associated inequalities. The solution of this problem requires the evaluation of four
constants. Two of them were found by G. Bennett. We find one of the two unknown constants and also prove one optimal
weighted-type estimate regarding the remaining constant.
Норми Чезаро i Копсона невiд’ємних послiдовностей визначаються як lp -норми їхнiх арифметичних середнiх i
вiдповiдних спряжених середнiх. Вiдомо, що для 1 < p < \infty цi норми еквiвалентнi. У 1996 р. Г. Беннетт поставив
задачу про знаходження найкращих сталих у нерiвностях, що описують цю еквiвалентнiсть. Розв’язок цiєї задачi
вимагає оцiнок чотирьох сталих. Двi з них були знайденi Г. Беннеттом. У цiй статтi знайдено одну з двох невiдомих
сталих. Доведено також оптимальну оцiнку вагового типу для сталої, що залишилася.
1. Introduction. Let 1 < p < \infty . Denote by \mathrm{c}\mathrm{e}\mathrm{s}(p) the set of all sequences \bfx = \{ xn\} such that
\| \bfx \| ces(p) =
\Biggl( \infty \sum
n=1
\Biggl(
1
n
n\sum
k=1
| xk|
\Biggr) p\Biggr) 1/p
< \infty .
By Hardy’s inequality [4] (Ch. 9), lp \subset \mathrm{c}\mathrm{e}\mathrm{s}(p), 1 < p < \infty .
We consider also the space \mathrm{c}\mathrm{o}\mathrm{p}(p) which is defined as the set of all sequences \bfx = \{ xn\} such
that
\| \bfx \| cop(p) =
\Biggl( \infty \sum
n=1
\Biggl( \infty \sum
k=n
| xk|
k
\Biggr) p\Biggr) 1/p
< \infty .
For any 1 < p < \infty , \mathrm{c}\mathrm{e}\mathrm{s}(p) = \mathrm{c}\mathrm{o}\mathrm{p}(p) (see [1], \S 10). Moreover, G. Bennett [1] proved the following
theorem.
Theorem 1.1. If p \geq 2, then
\| \bfx \| ces(p) \leq \zeta (p)1/p\| \bfx \| cop(p), (1.1)
where
\zeta (p) =
\infty \sum
n=1
1
np
, 1 < p < \infty .
If 1 < p \leq 2, then
\| \bfx \| cop(p) \leq (p - 1)1/p\| \bfx \| ces(p). (1.2)
The constants are both best possible.
c\bigcirc V. I. KOLYADA, 2019
220 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
ON CÈSARO AND COPSON NORMS OF NONNEGATIVE SEQUENCES 221
Furthermore, G. Bennett [1] posed the problem: find the best constants in the inequalities
\| \bfx \| ces(p) \leq Ap\| \bfx \| cop(p) for 1 < p < 2 (1.3)
and
\| \bfx \| cop(p) \leq Bp\| \bfx \| ces(p) for p > 2. (1.4)
Similar relationships between Lp-norms of the Hardy operator and its dual for functions on
(0,+\infty ) were studied in the work [5].
Denote by \scrM +(\BbbR +) the class of all nonnegative measurable functions on \BbbR + \equiv (0,+\infty ). Let
f \in \scrM +(\BbbR +). Set
Hf(x) =
1
x
x\int
0
f(t) dt and H\ast f(x) =
\infty \int
x
f(t)
t
dt.
By Hardy’s inequalities [4] (Ch. 9), these operators are bounded in Lp(\BbbR +) for any 1 < p < \infty .
Furthermore, it is easy to show that for any 1 < p < \infty the Lp-norms of Hf and H\ast f are
equivalent.
The main result in [5] is the following theorem.
Theorem 1.2. Let f \in \scrM +(\BbbR +) and let 1 < p < \infty . Then
(p - 1)\| Hf\| p \leq \| H\ast f\| p \leq (p - 1)1/p\| Hf\| p (1.5)
if 1 < p \leq 2, and
(p - 1)1/p\| Hf\| p \leq \| H\ast f\| p \leq (p - 1)\| Hf\| p (1.6)
if 2 \leq p < \infty . All constants in (1.5) and (1.6) are the best possible.
As it was observed in [5], the first inequality in (1.6) can be derived from the results obtained
in [3] and [6].
We observe also that the first inequality in (1.6) and the second inequality in (1.5) were obtained
in [1] (\S 21). We didn’t mention this fact in the paper [5] because we learned about the monograph
[1] when [5] was already published.
Note that the constant in the first inequality in (1.6) differs from that in (1.1).
One of the main results of this paper is that the best constant in (1.4) is Bp = p - 1. Our proof
of this result doesn’t rely on the second inequality in (1.6) (apparently it cannot be directly derived
from the latter inequality).
As for the best constant in the inequality (1.3), this problem remains open. However, we prove
the following result: if 1 < p \leq 2, then
\infty \sum
n=1
\Biggl(
1
n
n\sum
k=1
xk
\Biggr) p
\leq (p - 1) - (p - 1)
\infty \sum
n=1
\Biggl( \infty \sum
k=n
xk
k
wp(k)
\Biggr) p
, (1.7)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
222 V. I. KOLYADA
where
wp(n) =
\Biggl(
np - 1
\infty \sum
k=n
1
kp
\Biggr) 1/p
.
The constant in (1.7) is optimal.
Since wp(n) \leq \zeta (p)1/p (see Lemma 2.1 below), (1.7) implies that
\| \bfx \| ces(p) \leq
\zeta (p)1/p
(p - 1)1/p\prime
\| \bfx \| cop(p), 1 < p \leq 2. (1.8)
This result agrees with (1.1) for p = 2. Besides, the constant in (1.8) has the optimal asymptotic
behaviour as p \rightarrow 1 + . However, it is not difficult to show that for 1 < p < 2 the value of the
constant in (1.8) is not the best possible.
We observe that the main role in this paper belongs to Lemma 2.3 which gives explicit link
between Cèsaro and Copson norms.
2. Lemmas. The following lemma was proved in [1, p. 14].
Lemma 2.1. Let 1 < p < \infty . Set
\nu p(n) = np - 1
\infty \sum
k=n
1
kp
, n \in \BbbN . (2.1)
Then
\bigl\{
\nu p(n)
\bigr\}
strictly decreases as n increases and
1
p - 1
< \nu p(n) \leq \zeta (p), n \in \BbbN . (2.2)
The constants in (2.2) are both best possible and there is equality on the right only when n = 1.
Remark 2.1. We observe that the decrease of the sequence
\bigl\{
\nu p(n)
\bigr\}
was stated in [1] without
proof. However, the proof follows immediately from the well-known representation
\infty \sum
k=n
1
kp
=
1
\Gamma (p)
\infty \int
0
tp - 1e - nt
1 - e - t
dt. (2.3)
Indeed, from (2.3) we have
\nu p(n) =
1
\Gamma (p)
\infty \int
0
zp - 1e - z
n(1 - e - z/n)
dz.
It remains to note that the function \varphi (x) = x/(1 - e - x), x > 0, increases on (0,\infty ).
Observe also that
\infty \sum
k=n+1
1
kp
\leq
\infty \int
n
dx
xp
=
1
(p - 1)np - 1
. (2.4)
Together with the left inequality in (2.2), this implies that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\nu p(n) =
1
p - 1
. (2.5)
As usual, we set p\prime = p/(p - 1) for 1 < p < \infty .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
ON CÈSARO AND COPSON NORMS OF NONNEGATIVE SEQUENCES 223
Lemma 2.2. Let 1 < p < \infty and let N \geq 2 be an integer number. Set
x(N)
n =
\left\{ n - 1/p for n = 1, . . . , N,
0 for n > N
and \bfx (N) =
\bigl\{
x
(N)
n
\bigr\}
. Then
\mathrm{l}\mathrm{i}\mathrm{m}
N\rightarrow \infty
\bigm\| \bigm\| \bfx (N)
\bigm\| \bigm\|
ces(p)
\mathrm{l}\mathrm{n}N
= p\prime (2.6)
and
\mathrm{l}\mathrm{i}\mathrm{m}
N\rightarrow \infty
\bigm\| \bigm\| \bfx (N)
\bigm\| \bigm\|
cop(p)
\mathrm{l}\mathrm{n}N
= p. (2.7)
Proof. For the sum s
(N)
n =
\sum n
k=1
x
(N)
k we have
p\prime
\bigl[
(n+ 1)1/p
\prime - 1
\bigr]
\leq s(N)
n \leq p\prime n1/p\prime , 1 \leq n \leq N,
and s
(N)
n = s
(N)
N for n > N. It easily follows that
(p\prime )p(1 - \varepsilon N ) \mathrm{l}\mathrm{n}N \leq
\infty \sum
n=1
\Biggl(
s
(N)
n
n
\Biggr) p
\leq (p\prime )p(\mathrm{l}\mathrm{n}N + p\prime ),
where \varepsilon N > 0, \varepsilon N \rightarrow 0 as N \rightarrow \infty . These inequalities imply (2.6).
Further, for 2 \leq n \leq N,
\eta (N)
n =
\infty \sum
k=n
x
(N)
k
k
=
N\sum
k=n
k - 1/p - 1 \leq
N\int
n - 1
dx
x1+1/p
\leq p
(n - 1)1/p
.
Thus,
\infty \sum
n=1
\bigl(
\eta (N)
n
\bigr) p \leq pp \mathrm{l}\mathrm{n}N + C,
where C is a constant. On the other hand,
\eta (N)
n \geq p
\bigl(
n - 1/p - (N + 1) - 1/p
\bigr)
, 1 \leq n \leq N.
From here, for any \varepsilon > 0 and sufficiently big N,
N\sum
n=1
\bigl(
\eta (N)
n
\bigr) p \geq pp(1 - \varepsilon ) \mathrm{l}\mathrm{n}N.
These estimates imply (2.7).
Lemma 2.2 is proved.
Now we prove our main lemma. Throughout this paper, for a nonnegative sequence \{ xn\} , we
denote
sn =
n\sum
k=1
xk, \xi n =
xn
n
, \eta n =
\infty \sum
k=n
\xi k. (2.8)
Also, we use notation (2.1).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
224 V. I. KOLYADA
Lemma 2.3. Let \{ xn\} be a positive sequence and let 1 < p < \infty . Set
\alpha n =
1\int
0
\Bigl( sn
n
- y\xi n
\Bigr) p - 1
dy,
\beta n =
1\int
0
(\eta n+1 + y\xi n)
p - 1 dy,
and
\gamma n =
1\int
0
\Bigl( sn
n
- y\xi n
\Bigr)
(\eta n+1 + y\xi n)
p - 2 dy.
Then
\infty \sum
n=1
\Bigl( sn
n
\Bigr) p
= p
\infty \sum
n=1
xn\alpha n\nu p(n) (2.9)
and
\infty \sum
n=1
\eta pn = p
\infty \sum
n=1
xn\beta n = p(p - 1)
\infty \sum
n=1
xn\gamma n. (2.10)
Proof. First, applying summation by parts, we have (s0 = 0)
Ip =
\infty \sum
n=1
\Bigl( sn
n
\Bigr) p
=
\infty \sum
n=1
(spn - spn - 1)
\infty \sum
k=n
1
kp
=
= p
\infty \sum
n=1
\nu p(n)
np - 1
xn
1\int
0
(sn - yxn)
p - 1dy = p
\infty \sum
n=1
xn\alpha n\nu p(n),
which gives (2.9).
Further,
Jp =
\infty \sum
n=1
\eta pn =
\infty \sum
n=1
n(\eta pn - \eta pn+1) = p
\infty \sum
n=1
xn
1\int
0
(\eta n+1 + y\xi n)
p - 1 dy, (2.11)
and we obtain the left equality in (2.10).
Next, we apply summation by parts one more time. Set \varphi n(y) = \eta n+1 + y\xi n, y \in [0, 1]. Then
\varphi n(y)
p - 1 - \varphi n+1(y)
p - 1 = (\eta n+2 + \xi n+1 + y\xi n)
p - 1 - (\eta n+2 + y\xi n+1)
p - 1 =
= (p - 1)
\xi n+1+y\xi n\int
y\xi n+1
(\eta n+2 + u)p - 2 du.
Thus,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
ON CÈSARO AND COPSON NORMS OF NONNEGATIVE SEQUENCES 225
1\int
0
\bigl[
\varphi n(y)
p - 1 - \varphi n+1(y)
p - 1
\bigr]
dy =
= (p - 1)
\left[ 1\int
0
\xi n+1\int
y\xi n+1
(\eta n+2 + u)p - 2 dudy +
1\int
0
y\xi n\int
0
(\eta n+1 + u)p - 2 dudy
\right] =
= (p - 1)
\left[ \xi n+1
1\int
0
y(\eta n+2 + y\xi n+1)
p - 2 dy + \xi n
1\int
0
(1 - y)(\eta n+1 + y\xi n)
p - 2 dy
\right] .
Using this equality and applying summation by parts in (2.11), we obtain
Jp = p(p - 1)
\infty \sum
n=1
sn
\left[ \xi n+1
1\int
0
y(\eta n+2 + y\xi n+1)
p - 2dy + \xi n
1\int
0
(1 - y)(\eta n+1 + y\xi n)
p - 2dy
\right] .
As above, we assume that s0 = 0. We have
y
\infty \sum
n=1
\xi n+1(\eta n+2 + y\xi n+1)
p - 2sn + (1 - y)
\infty \sum
n=1
\xi n(\eta n+1 + y\xi n)
p - 2sn =
= y
\infty \sum
n=1
\xi n(\eta n+1 + y\xi n)
p - 2sn - 1 + (1 - y)
\infty \sum
n=1
\xi n(\eta n+1 + y\xi n)
p - 2sn =
=
\infty \sum
n=1
\xi n(sn - yxn)(\eta n+1 + y\xi n)
p - 2.
Thus,
Jp = p(p - 1)
\infty \sum
n=1
xn
1\int
0
\Bigl( sn
n
- y\xi n
\Bigr)
(\eta n+1 + y\xi n)
p - 2 dy,
which is the right equality in (2.10).
Lemma 2.3 is proved.
3. Main results. As above, we use notations (2.1) and (2.8). First we obtain the optimal constant
in the inequality (1.4).
Theorem 3.1. Let \bfx = \{ xn\} be a nonnegative sequence and let 2 \leq p < \infty . Then
\| \bfx \| cop(p) \leq (p - 1)\| \bfx \| ces(p). (3.1)
The constant is optimal.
Proof. We shall use notations introduced in Lemma 2.3. First we observe that by Hölder’s
inequality
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
226 V. I. KOLYADA
\gamma n =
1\int
0
\Bigl( sn
n
- y\xi n
\Bigr)
(\eta n+1 + y\xi n)
p - 2 dy \leq
\leq
\left( 1\int
0
\Bigl( sn
n
- y\xi n
\Bigr) p - 1
dy
\right) 1/(p - 1)\left( 1\int
0
(\eta n+1 + y\xi n)
p - 1dy
\right) (p - 2)/(p - 1)
=
= \alpha 1/(p - 1)
n \beta (p - 2)/(p - 1)
n .
Using the second of equalities (2.10), and applying again the Hölder inequality, we get
Jp =
\infty \sum
n=1
\eta pn = p(p - 1)
\infty \sum
n=1
xn\gamma n \leq p(p - 1)
\infty \sum
n=1
xn\alpha
1/(p - 1)
n \beta (p - 2)/(p - 1)
n \leq
\leq (p - 1)
\Biggl(
p
\infty \sum
n=1
xn\alpha n
\Biggr) 1/(p - 1)\Biggl(
p
\infty \sum
n=1
xn\beta n
\Biggr) (p - 2)/(p - 1)
.
We observe that by (2.2)
\infty \sum
n=1
xn\alpha n \leq (p - 1)
\infty \sum
n=1
xn\alpha n\nu p(n). (3.2)
As above, set Ip =
\sum \infty
n=1
(sn/n)
p. Using (3.2), (2.9), and the first equality in (2.10), we have
Jp \leq (p - 1)p
\prime
I1/(p - 1)
p J (p - 2)/(p - 1)
p .
From here, Jp \leq (p - 1)pIp, which yields (3.1). It follows immediately from Lemma 2.2 that the
constant in (3.1) is the best possible.
Remark 3.1. It is clear that inequality (3.2) is strict except the case when xn = 0 for all n \in \BbbN .
Thus, the equality in (3.1) holds if and only if \bfx = \bfzero .
Applying Lemma 2.3, we obtain also the following result.
Theorem 3.2. Let \{ xn\} be a nonnegative sequence and let 1 < p \leq 2. Set wp(n) = \nu p(n)
1/p.
Then \Biggl( \infty \sum
n=1
\Biggl(
1
n
n\sum
k=1
xk
\Biggr) p\Biggr) 1/p
\leq
\leq (p - 1) - 1/p\prime
\Biggl( \infty \sum
n=1
\Biggl( \infty \sum
k=n
xk
k
wp(k)
\Biggr) p\Biggr) 1/p
. (3.3)
The constant is optimal.
Proof. We keep notations (2.8). Also, we set
\~xn = xnwp(n), \~sn =
n\sum
k=1
\~xk, \~\xi n =
\~xn
n
, and \~\eta n =
\infty \sum
k=n
\~\xi k.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
ON CÈSARO AND COPSON NORMS OF NONNEGATIVE SEQUENCES 227
Since the sequence
\bigl\{
\nu p(n)
\bigr\}
is decreasing, we have \~sn \geq wp(n)sn. Thus, applying Hölder’s inequa-
lity with the exponent p - 1 \in (0, 1], we obtain
\~\gamma n =
1\int
0
\biggl(
\~sn
n
- y\~\xi n
\biggr)
(\~\eta n+1 + y\~\xi n)
p - 2dy \geq
\geq wp(n)
\left( 1\int
0
\Bigl( sn
n
- y\xi n
\Bigr) p - 1
dy
\right) 1/(p - 1)\left( 1\int
0
(\~\eta n+1 + y\~\xi n)
p - 1dy
\right) (p - 2)/(p - 1)
.
As above, we denote
\alpha n =
1\int
0
\Bigl( sn
n
- y\xi n
\Bigr) p - 1
dy.
Further, set
\~\beta n =
1\int
0
\bigl(
\~\eta n+1 + y\~\xi n
\bigr) p - 1
dy.
Applying the right equality in (2.10) to the sequence \{ \~xn\} , we have
\~Jp =
\infty \sum
n=1
\~\eta pn = p(p - 1)
\infty \sum
n=1
\~xn\~\gamma n.
Using estimate for \~\gamma n obtained above, and applying again Hölder’s inequality, we get
\~Jp = p(p - 1)
\infty \sum
n=1
\~xn\~\gamma n \geq p(p - 1)
\infty \sum
n=1
(\nu p(n)xn\alpha n)
1/(p - 1)(\~xn \~\beta n)
(p - 2)/(p - 1) \geq
\geq (p - 1)
\Biggl(
p
\infty \sum
n=1
xn\alpha n\nu p(n)
\Biggr) 1/(p - 1)\Biggl(
p
\infty \sum
n=1
\~xn \~\beta n
\Biggr) (p - 2)/(p - 1)
.
Now we apply the left equality in (2.10) to \{ \~xn\} and equality (2.9) to \{ xn\} . This gives that
\~Jp \geq (p - 1)I1/(p - 1)
p
\~J (p - 2)/(p - 1)
p ,
where
Ip =
\infty \sum
n=1
\Bigl( sn
n
\Bigr) p
.
The latter inequality implies (3.3).
Let now \bfx (N) =
\bigl\{
x
(N)
n
\bigr\}
be the sequence defined in Lemma 2.2. By (2.6),
\mathrm{l}\mathrm{i}\mathrm{m}
N\rightarrow \infty
\| \bfx (N)\| ces(p)
\mathrm{l}\mathrm{n}N
=
p
p - 1
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
228 V. I. KOLYADA
Further, set \~x(N)
n = x
(N)
n wp(n) and \~\bfx (N) =
\bigl\{
\~x
(N)
n
\bigr\}
. It easily follows from (2.5) and (2.7) that
\mathrm{l}\mathrm{i}\mathrm{m}
N\rightarrow \infty
\| \~\bfx (N)\| cop(p)
\mathrm{l}\mathrm{n}N
=
p
(p - 1)1/p
.
It shows that the constant in (3.3) cannot be improved.
Theorem 3.2 is proved.
Remark 3.2. As we have mentioned above, we do not know the value of the best constant Ap
in (1.3). By Lemma 2.2,
Ap \geq
1
p - 1
. (3.4)
On the other hand, we have
sn =
n\sum
k=1
k(\eta k - \eta k+1) \leq
n\sum
k=1
\eta k (3.5)
and by Hardy’s inequality\Biggl( \infty \sum
n=1
\Bigl( sn
n
\Bigr) p\Biggr) 1/p
\leq
\Biggl( \infty \sum
n=1
\Biggl(
1
n
n\sum
k=1
\eta k
\Biggr) p\Biggr) 1/p
\leq p\prime
\Biggl( \infty \sum
n=1
\eta pn
\Biggr) 1/p
.
Thus,
Ap \leq p\prime . (3.6)
It follows from (3.4) and (3.6) that
\mathrm{l}\mathrm{i}\mathrm{m}
p\rightarrow 1
(p - 1)Ap = 1. (3.7)
Clearly, estimate (3.6) is too rough if p \rightarrow 2. Indeed, by (1.1), the best constant A2 = \zeta (2) =
= \pi 2/6 < 2.
Another upper bound for Ap can be derived from Theorem 3.2. Since \nu p(n) decreases (see
Lemma 2.1), it follows from (3.3) that for 1 < p \leq 2
\infty \sum
n=1
\Bigl( sn
n
\Bigr) p
\leq 1
(p - 1)p - 1
\infty \sum
n=1
\nu p(n)\eta
p
n. (3.8)
Further, by (2.2), we have
\infty \sum
n=1
\Bigl( sn
n
\Bigr) p
\leq \zeta (p)
(p - 1)p - 1
\infty \sum
n=1
\eta pn, 1 < p \leq 2. (3.9)
For p = 2 this inequality coincides with (1.1). Furthermore, if \~Ap = \zeta (p)1/p(p - 1) - 1/p\prime , then
\mathrm{l}\mathrm{i}\mathrm{m}
p\rightarrow 1
(p - 1) \~Ap = 1,
which agrees with (3.7). However, for 1 < p < 2 the constant in (3.9) is not optimal. It can be easily
shown, using (3.8) and (3.5).
Finally, concerning the best constant Ap in (1.3), we can only state that it satisfies the inequalities
1
p - 1
\leq Ap \leq
\zeta (p)1/p
(p - 1)1/p\prime
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
ON CÈSARO AND COPSON NORMS OF NONNEGATIVE SEQUENCES 229
References
1. Bennett G. Factorizing the classical inequalities // Mem. Amer. Math. Soc. – 1996. – 120, № 576.
2. Bennett C., Sharpley R. Interpolation of operators. – Boston: Acad. Press, 1988.
3. Boza S., Soria J. Solution to a conjecture on the norm of the Hardy operator minus the identity // J. Funct. Anal. –
2011. – 260. – P. 1020 – 1028.
4. Hardy G. H., Littlewood J. E., Pólya G. Inequalities. – 2nd ed. – Cambridge: Cambridge Univ. Press, 1967.
5. Kolyada V. I. Optimal relationships between Lp -norms for the Hardy operator and its dual // Ann. Mat. Pura ed
Appl. – 2014. – 4, № 2. – P. 423 – 430.
6. Kruglyak N., Setterqvist E. Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions //
Proc. Amer. Math. Soc. – 2008. – 136. – P. 2005 – 2013.
Received 26.09.18
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
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| id | umjimathkievua-article-1433 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:17Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ac/046de045f132c1ab57f32eeb9d05b7ac.pdf |
| spelling | umjimathkievua-article-14332019-12-05T08:54:43Z On Cèsaro and Copson norms of nonnegative sequences Про норми чезаро i Копсона невiд’ємних послiдовностей Kolyada, V. I. Коляда, В. І. The C`esaro and Copson norms of a nonnegative sequence are lp-norms of its arithmetic means and the corresponding conjugate means. It is well known that, for $1 < p < \infty$, these norms are equivalent. In 1996, G. Bennett posed the problem of finding the best constants in the associated inequalities. The solution of this problem requires the evaluation of four constants. Two of them were found by G. Bennett. We find one of the two unknown constants and also prove one optimal weighted-type estimate regarding the remaining constant. Норми Чезаро i Копсона невiд’ємних послiдовностей визначаються як lp-норми їхнiх арифметичних середнiх i вiдповiдних спряжених середнiх. Вiдомо, що для $1 < p < \infty$ цi норми еквiвалентнi. У 1996 р. Г. Беннетт поставив задачу про знаходження найкращих сталих у нерiвностях, що описують цю еквiвалентнiсть. Розв’язок цiєї задачi вимагає оцiнок чотирьох сталих. Двi з них були знайденi Г. Беннеттом. У цiй статтi знайдено одну з двох невiдомих сталих. Доведено також оптимальну оцiнку вагового типу для сталої, що залишилася. Institute of Mathematics, NAS of Ukraine 2019-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1433 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 2 (2019); 220-229 Український математичний журнал; Том 71 № 2 (2019); 220-229 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1433/417 Copyright (c) 2019 Kolyada V. I. |
| spellingShingle | Kolyada, V. I. Коляда, В. І. On Cèsaro and Copson norms of nonnegative sequences |
| title | On Cèsaro and Copson norms of nonnegative sequences |
| title_alt | Про норми чезаро i Копсона невiд’ємних послiдовностей |
| title_full | On Cèsaro and Copson norms of nonnegative sequences |
| title_fullStr | On Cèsaro and Copson norms of nonnegative sequences |
| title_full_unstemmed | On Cèsaro and Copson norms of nonnegative sequences |
| title_short | On Cèsaro and Copson norms of nonnegative sequences |
| title_sort | on cèsaro and copson norms of nonnegative sequences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1433 |
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