On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case
Motivated by the results of Huang, we deduce a pair of discrete multidimensional Hilbert-type inequalities involving a homogeneous kernel of negative degree. We also establish conditions under which the constant factors involved in the established inequalities are the best possible. Finally, we cons...
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| author | Krnić, M. Vuković, P. Крнич, М. Вукович, П. |
| author_facet | Krnić, M. Vuković, P. Крнич, М. Вукович, П. |
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| description | Motivated by the results of Huang, we deduce a pair of discrete multidimensional Hilbert-type inequalities involving a homogeneous kernel of negative degree. We also establish conditions under which the constant factors involved in the established inequalities are the best possible. Finally, we consider some particular settings with homogeneous kernels and weight functions. In this way, we obtain generalizations of some results known from the literature. |
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UDC 517.5
M. Krnić, P. Vuković (Univ. Zagreb, Croatia)
ON SOME MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES
IN A DISCRETE CASE*
ПРО ДЕЯКI БАГАТОВИМIРНI НЕРIВНОСТI ГIЛЬБЕРТОВОГО ТИПУ
У ДИСКРЕТНОМУ ВИПАДКУ
Motivated by results of Huang, we derive a pair of discrete multidimensional Hilbert-type inequalities involving a
homogeneous kernel of negative degree. We also establish conditions under which the constant factors involved in the
established inequalities are the best possible. Finally, we consider some particular settings with homogeneous kernels and
weight functions. In such a way we obtain generalizations of some results known from the literature.
З метою узагальнення результатiв Хуанга отримано двi дискретнi багатовимiрнi нерiвностi гiльбертового типу з
однорiдним ядром вiд’ємного степеня. Також встановлено умови, за яких сталi множники, що входять до отриманих
нерiвностей, є найкращими з можливих. Розглянуто деякi конкретнi випадки однорiдних ядер та вагових функцiй.
Це дає змогу узагальнити деякi вiдомi результати.
1. Introduction. Hilbert’s inequality is one of the most significant weighted inequalities in mathe-
matical analysis and its applications. Through the years, Hilbert-type inequalities were discussed by
numerous authors, who either reproved them using various techniques, or applied and generalized
them in many different ways. For more details about Hilbert’s inequality the reader is referred to [1]
or [3].
Although classical, Hilbert’s inequality is still of interest to numerous mathematicians. In this
paper we refer to the recent paper [2], where Q. Huang obtained multidimensional discrete Hilbert-
type inequality equipped with conjugate parameters. His result is contained in the following theorem.
Theorem 1.1. Suppose that n ∈ N \ {1}, pi, ri > 1, i = 1, . . . , n,
∑n
i=1
1
pi
=
∑n
i=1
1
ri
= 1,
1
qn
= 1 − 1
pn
, λ > 0, 0 < α < 2, β ≥ −1
2
, λαmax
{
1
2− α
, 1
}
≤ min1≤i≤n{ri}, a(i)
mi ≥ 0,
mi ∈ N, such that
0 <
∞∑
mi=1
(mi + β)pi(1−λα/ri)−1
(
a(i)
mi
)pi
<∞, i = 1, . . . , n.
Then the following two inequalities hold and are equivalent:
∞∑
mn=1
. . .
∞∑
m1=1
1[∑n
i=1
(mi + β)α
]λ n∏
i=1
a(i)
mi
<
<
α1−n
Γ(λ)
n∏
i=1
Γ
(
λ
ri
)( ∞∑
mi=1
(mi + β)
pi
(
1−λαri
)
−1
(
a(i)
mi
)pi)1/pi
,
* This research was supported by the Croatian Ministry of Science, Education, and Sports, under Research Grants
036-1170889-1054 (first author) and 058-1170889-1050 (second author).
c© M. KRNIĆ, P. VUKOVIĆ, 2013
802 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
ON SOME MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES IN A DISCRETE CASE 803
∞∑
mn=1
(mn + β)
λαqn
rn
−1
∞∑
mn−1=1
. . .
∞∑
m1=1
∏n−1
i=1
a(i)
mi[∑n
i=1
(mi + β)α
]λ
qn
1/qn
<
<
Γ (λ/rn)
αn−1Γ(λ)
n−1∏
i=1
Γ
(
λ
ri
)( ∞∑
mi=1
(mi + β)
pi
(
1−λαri
)
−1
(
a(i)
mi
)pi)1/pi
.
The constant factor
α1−n
Γ(λ)
∏n
i=1
Γ
(
λ
ri
)
is the best possible.
In the previous theorem Γ denotes the usual Gamma function. Besides, the best possible constant
factor means that it can not be replaced with a smaller constant, so that the appropriate inequality
still holds.
On the other hand Yang et al. [6], obtained the result which provides an unified treatment of
multidimensional Hilbert-type inequality in the setting with conjugate exponents. All the measures
are assumed to be σ-finite on measure space Ω.
Theorem 1.2. Let n ≥ 2 be an integer and let p1, . . . , pn be conjugate parameters such that
pi > 1, i = 1, . . . , n. Let K : Ωn → R and φi,j : Ω→ R, i, j = 1, . . . , n, be nonnegative measurable
functions such that
∏n
i,j=1
φij(xj) = 1. Then the following inequalities hold and are equivalent:
∫
Ωn
K(x1, . . . , xn)
n∏
i=1
fi(xi)dµ1(x1) . . . dµn(xn) ≤
≤
n∏
i=1
∫
Ω
Fi(xi)f
pi
i (xi)φ
pi
ii (xi)dµi(xi)
1/pi
(1.1)
and ∫
Ω
h(xn)
∫
Ωn−1
K(x1, . . . , xn)
n−1∏
i=1
fi(xi)dµ1(x1) . . . dµn−1(xn−1)
qn dµn(xn) ≤
≤
n−1∏
i=1
∫
Ω
Fi(xi)f
pi
i (xi)φ
pi
ii (xi)dµi(xi)
qn/pi , (1.2)
where
Fi(xi) =
∫
Ωn−1
K(x1, . . . , xn)×
×
n∏
j=1,j 6=i
φpiij (xj)dµ1(x1) . . . dµi−1(xi−1)dµi+1(xi+1) . . . dµn(xn), i = 1, . . . , n,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
804 M. KRNIĆ, P. VUKOVIĆ
h(xn) = φ−qnnn (xn)F 1−qn
n (xn) and
1
qn
=
n−1∑
i=1
1
pi
.
In the literature, the inequalities related to (1.1) are usually referred to as the Hilbert-type in-
equalities, while the inequalities related to (1.2) are usually called Hardy – Hilbert-type inequalities.
For more details about their equivalence, the reader is referred to [6].
The main purpose of this paper is to generalize Theorem 1.1 in the view of Theorem 1.2. More
precisely, in the sequel we derive the discrete forms of inequalities (1.1) and (1.2) containing the
homogeneous kernel. Besides, considerable attention is dedicated to the investigation of the best
possible constant factors in obtained inequalities, which can be achieved in some general settings. As
an application, we also consider some particular settings of our general results which reduce to some
recent results, known from the literature.
The techniques that will be used in the proofs are mainly based on classical real analysis. Further,
throughout the whole paper, without further explanation, all the series and integrals are assumed to
be convergent.
2. Main results. We start this section with the application of Theorem 1.2, which will give
the discrete forms of inequalities (1.1) and (1.2). By using the notations as in the above mentioned
theorem, we consider the case where Ω = N, the measures µi, i = 1, . . . , n, are counting measures,
and the kernel K is the nonnegative homogeneous function of degree −λ, λ > 0.
In order to obtain the constant factors involved in the inequalities, we define the function
k (β1, . . . , βn−1) by
k (β1, . . . , βn−1) :=
∫
(0,∞)n−1
K(1, t1 . . . , tn−1)tβ11 . . . t
βn−1
n−1 dt1 . . . dtn−1, (2.1)
where we suppose that k (β1, . . . , βn−1) <∞ for β1, . . . , βn−1 > −1 and β1+. . .+βn−1+n < λ+1.
Further, let Aij , i, j = 1, . . . , n, be the real numbers satisfying
n∑
i=1
Aij = 0, j = 1, 2, . . . , n. (2.2)
We also define
αi =
n∑
j=1
Aij , i = 1, 2, . . . , n. (2.3)
Besides, we consider the discrete weighted functions involving real differentiable functions. More
precisely, we have the following definition.
Definition 2.1. Let r ∈ R. We denote by H(r) the set of all nonnegative differentiable functions
u : (0,∞)→ R satisfying the following conditions:
(1) u is strictly increasing on (0,∞) and there exists x0 ∈ (0,∞) such that u(x0) = 1;
(2) limx→∞ u(x) =∞, [u(x)]ru′(x) is decreasing on (0,∞).
Now, regarding the above notations and definitions, we are ready to state and prove our first
general result.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
ON SOME MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES IN A DISCRETE CASE 805
Theorem 2.1. Let p1, . . . , pn be conjugate parameters such that pi > 1, i = 1, . . . , n, and
let
1
qn
=
∑n−1
i=1
1
pi
. Let K : (0,∞)n → R be nonnegative homogeneous function of degree −λ,
λ > 0, strictly decreasing in each variable, and let Aij , i, j = 1, . . . , n, and αi, i = 1, . . . , n, be
real parameters satisfying (2.2) and (2.3). If a(i)
mi ≥ 0, mi ∈ N, and ui ∈ H(piAij), i, j = 1, . . . , n,
i 6= j, then we have the following equivalent inequalities:
∞∑
mn=1
. . .
∞∑
m1=1
K(u1(m1), . . . , un(mn))
n∏
i=1
a(i)
mi
≤
≤ L
n∏
i=1
( ∞∑
mi=1
[ui(mi)]
n−λ−1+piαi [u′i(mi)]
1−pi
(
a(i)
mi
)pi)1/pi
, (2.4)
∞∑
mn=1
[un(mn)](1−qn)(n−1−λ)−qnαn ×
×
∞∑
mn−1=1
. . .
∞∑
m1=1
K(u1(m1), . . . , un(mn))
n−1∏
i=1
a(i)
mi
qn1/qn
≤
≤ L
n−1∏
i=1
( ∞∑
mi=1
[ui(mi)]
n−λ−1+piαi [u′i(mi)]
1−pi
(
a(i)
mi
)pi)1/pi
, (2.5)
where
L = k(p1A12, . . . , p1A1n)1/p1k(λ− n− p2(α2 −A22), p2A23, . . . , p2A2n)1/p2 . . .
. . . k(pnAn2, . . . , pnAn,n−1, λ− n− pn(αn −Ann))1/pn , (2.6)
and piAij > −1, i 6= j, pi(Aii − αi) > n− λ− 1.
Proof. Rewrite the inequality (1.1) for the counting measure on N,
(φij ◦ uj)(mj) =
[
uj(mj)]
Aij [u′j(mj)
]1/pi , i 6= j,
(φii ◦ ui)(mi) =
[
ui(mi)]
Aii [u′i(mi)
]1/pi−1
,
and the sequences
(
a
(i)
mi
)
, i = 1, . . . , n. Obviously, these substitutions are well defined, since ui,
i = 1, . . . , n, are injective functions. Thus, in the above setting we have
∞∑
mn=1
. . .
∞∑
m1=1
K
(
u1(m1), . . . , un(mn)
) n∏
i=1
a(i)
mi
≤
≤
n∏
i=1
( ∞∑
mi=1
[
ui(mi)
]piAii
[
u′i(mi)
]1−pi(F ◦ ui)(mi)
(
a(i)
mi
)pi)1/pi
, (2.7)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
806 M. KRNIĆ, P. VUKOVIĆ
where
(F ◦ ui)(mi) =
∞∑
mn=1
. . .
∞∑
mi+1=1
∞∑
mi−1=1
∞∑
m1=1
K
(
u1(m1), . . . , un(mn)
)
×
×
n∏
j=1,j 6=i
[uj(mj)]
piAiju′j(mj)
.
Our next task is to estimate the functions (F ◦ ui)(mi), i = 1, . . . , n. Since the kernel K is
strictly decreasing in each variable and ui ∈ H(piAij), i 6= j, we conclude that the functions Fi ◦ui,
i = 1, . . . , n, are strictly decreasing. Hence, we have
(F1 ◦ u1)(m1) ≤
∫
(0,∞)n−1
K
(
u1(m1), u2(x2), . . . , un(xn)
)
×
×
n∏
j=2
(
[uj(xj)]
p1A1ju′j(xj)
)
dx2 . . . dxn, (2.8)
since the left-hand side of this inequality is obviously the lower Darboux sum for the integral on the
right-hand side of inequality. Further, by using the substitution ti = ui(xi), i = 2, . . . , n, from (2)
we get
(F1 ◦ u1)(m1) ≤
∫
(0,∞)n−1
K(u1(m1), t2, . . . , tn)
n∏
j=2
t
p1A1j
j dt2 . . . dtn,
wherefrom by using the homogeneity of the kernel K and the obvious change of variables, we have
(F1 ◦ u1)(m1) ≤
∫
(0,∞)n−1
[
u1(m1)
]−λ
K
(
1, t2/u1(m1), . . . , tn/u1(m1)
)
×
×
n∏
j=2
t
p1A1j
j dt2 . . . dtn =
=
[
u1(m1)
]n−1−λ+p1
(
α1−A11)
k(p1A12, . . . , p1A1n
)
.
By using the same arguments as for the function F1 ◦ u1, we also get
(F2 ◦ u2)(m2) ≤
∫
(0,∞)n−1
K(t1, u2(m2), t3, . . . , tn)
n∏
j=1,j 6=2
t
p2A2j
j dt1dt3 . . . dtn. (2.9)
Now, let J denotes the right-hand side of the inequality (2.9). It is easy to see that the transformation
of variables
t1 = u2(m2)
1
v2
, ti = u2(m2)
vi
v2
, i = 3, . . . , n,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
ON SOME MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES IN A DISCRETE CASE 807
yields
∂(t1, t3, . . . , tn)
∂(v2, v3, . . . , vn)
=
[
u2(m2)
]n−1
v−n2 ,
where
∂(t1, t3, . . . , tn)
∂(v2, v3, . . . , vn)
denotes the Jacobian of the transformation.
Now, by using the homogeneity of the kernel K and the above change of variables, we have
J =
∫
(0,∞)n−1
t−λ1 K
(
1, u2(m2)/t1, t3/t1 . . . , tn/t1
) n∏
j=1,j 6=2
t
p2A2j
j dt1dt3 . . . dtn =
=
∫
(0,∞)n−1
[u2(m2)]−λvλ2K(1, v2, . . . , vn)[u2(m2)]p2(α2−A22)×
×v−p2(α2−A22)
2 vp2A23
2 . . . vp2A2n
n [u2(m2)]n−1v−n2 dv2dv3 . . . dvn =
= [u2(m2)]n−1−λ+p2(α2−A22)
∫
(0,∞)n−1
v
λ−n−p2(α2−A22)
2
n∏
j=3
v
p2A2j
j dv2 . . . dvn =
= [u2(m2)]n−1−λ+p2(α2−A22)k(λ− n− p2(α2 −A22), p2A23, . . . , p2A2n).
Hence, inequality (2.9) and the above equality yield
(F2 ◦ u2)(m2) ≤ [u2(m2)]n−1−λ+p2(α2−A22)k
(
λ− n− p2(α2 −A22), p2A23, . . . , p2A2n
)
.
In a similar manner we obtain
(Fi ◦ ui)(mi) ≤ [ui(mi)]
n−1−λ+pi(αi−Aii)×
×k
(
piAi2, . . . , piAi,i−1, s− n− pi(αi −Aii), piAi,i+1, . . . , piAin
)
,
for i = 3, . . . , n. This completes the proof of inequality (2.4).
The proof of the inequality (2.5) follows from the inequality (1.2), by using the same estimates
as in the first part of the proof.
The next problem we are dealing with in this section is to determine the conditions under which
the constant factor L, defined by (2.6), is the best possible in inequalities (2.4) and (2.5). Considering
Theorem 1.1, we see that the appropriate constant factor does not include any exponent. Bearing in
mind that fact, we shall find the conditions under which the constant L reduces to the form without
any exponents.
In order to obtain the constant factor without exponents, it is natural to impose the following
conditions on the parameters Aij :
pjAji = λ− n− pi(αi −Aii), i, j = 1, 2, . . . , n, i 6= j. (2.10)
If the parameters Aij satisfy the set of conditions (2.10), then the constant L from Theorem 2.1
reduces to the form
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
808 M. KRNIĆ, P. VUKOVIĆ
L∗ = k(Ã2, . . . , Ãn), (2.11)
where we use the abbreviations
Ãi = pjAji, i, j = 1, 2, . . . , n, i 6= j. (2.12)
Regarding the set of conditions (2.10), it is easy to see that the parameters Ãi satisfy the relation
n∑
i=1
Ãi = λ− n. (2.13)
Furthermore, by using (2.2) and (2.12), we have the following relationship between the parameters
Aii and Ãi, i = 1, 2, . . . , n:
Aii = −A1i −A2i − . . .−Ai−1,i −Ai+1,i − . . .−Ani =
= −Ãi
p1
− Ãi
p2
− . . .− Ãi
pi−1
− Ãi
pi+1
− . . .− Ãi
pn
=
= Ãi
(
1
pi
− 1
)
. (2.14)
Now, taking into account the relations (2.11), (2.12), and (2.14), the inequalities (2.4) and (2.5)
with the parameters Aij , i, j = 1, 2, . . . , n, satisfying the set of conditions (2.10), become
∞∑
mn=1
. . .
∞∑
m1=1
K(u1(m1), . . . , un(mn))
n∏
i=1
a(i)
mi
≤
≤ L∗
n∏
i=1
( ∞∑
mi=1
[ui(mi)]
−1−piÃi [u′i(mi)]
1−pi
(
a(i)
mi
)pi)1/pi
, (2.15)
and ∞∑
mn=1
[un(mn)](1−qn)(−1−pnÃn) ×
×
∞∑
mn−1=1
. . .
∞∑
m1=1
K(u1(m1), . . . , un(mn))
n−1∏
i=1
a(i)
mi
qn1/qn
≤
≤ L∗
n−1∏
i=1
( ∞∑
mi=1
[ui(mi)]
−1−piÃi [u′i(mi)]
1−pi
(
a(i)
mi
)pi)1/pi
, (2.16)
where the constant factor L∗ is defined by (2.11).
Now we are ready to prove that the constant factor L∗ is the best possible in both inequalities
(2.15) and (2.16). That is the content of the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
ON SOME MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES IN A DISCRETE CASE 809
Theorem 2.2. If the parameters Aij , i, j = 1, . . . , n, satisfy the conditions (2.2) and (2.10),
then the constant factor L∗ is the best possible in both inequalities (2.15) and (2.16).
Proof. It is enough to show that the constant factor L∗ is the best possible in inequality (2.15),
since (2.15) and (2.16) are equivalent inequalities.
For that sake, we consider the real sequences ã(i)
mi = [ui(mi)]
Ãi−ε/pu′i(mi), where ε > 0 is
sufficiently small number. Since ui ∈ H(Ãi), i = 1, . . . , n, we may assume that ui is strictly
increasing on (0,∞) and that there exists x0 ∈ (0,∞) such that ui(x0) = 1.
Therefore, by considering integral sums, we have
1
ε
=
∞∫
1
[ui(x)]−1−εd[ui(x)] <
∞∑
mi=1
[ui(mi)]
−1−εu′i(mi) =
=
∞∑
mi=1
[ui(mi)]
−1−piÃi [u′i(mi)]
1−pi
(
ã(i)
mi
)pi
<
< ϑi(1) +
∞∫
1
[ui(x)]−1−εd[ui(x)] = ϑi(1) +
1
ε
,
where the function ϑi is defined by ϑi(x) = [ui(x)]−1−εu′i(x). In other words, the following relation
is valid:
∞∑
mi=1
[ui(mi)]
−1−piÃi [u′i(mi)]
1−pi
(
ã(i)
mi
)pi
=
1
ε
+O(1), i = 1, . . . , n. (2.17)
Now, let us suppose that there exists a positive constant M, smaller than L∗, such that the inequality
(2.15) is still valid, if we replace L∗ with M. Hence, if we insert relations (2.17) in inequality (2.15),
with the constant M instead of L∗, we get
Ĩ :=
∞∑
mn=1
. . .
∞∑
m1=1
K
(
u1(m1), . . . , un(mn)
) n∏
i=1
ã(i)
mi
<
1
ε
(M + o(1)). (2.18)
On the other hand, let us estimate the left-hand side of inequality (2.15). Namely, by inserting the
above defined sequences
(
ã
(i)
mi
)
mi∈N
in the left-hand side of inequality (2.15), we easily get the
inequality
Ĩ >
∞∫
1
[u1(x1)]Ã1−ε/p1
∞∫
1
. . .
∞∫
1
K(u1(x1), . . . , un(xn)) ×
×
n∏
i=2
[ui(xi)]
Ãi−ε/pid[u2(x2)] . . . d[un(xn)]
d[u1(x1)]. (2.19)
Further, let J denotes the right-hand side of the inequality (2.19). By using the substitution ti =
=
ui(xi)
u1(x1)
, i = 2, . . . , n, we find that
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
810 M. KRNIĆ, P. VUKOVIĆ
J =
∞∫
1
[u1(x1)]−1−ε
∞∫
1/u1(x1)
. . .
∞∫
1/u1(x1)
K(1, t2, . . . , tn)
n∏
i=2
ti
Ãi−ε/pidt2 . . . dtn
d[u1(x1)].
Now, considering the obtained expression for J, we easily get inequality
J ≥
∞∫
1
[u1(x1)]−1−ε
∫
(0,∞)n−1
K(1, t2, . . . , tn)
n∏
i=2
ti
Ãi−ε/pidt2 . . . dtn
d[u1(x1)]−
−
∞∫
1
[u1(x1)]−1−ε
n∑
j=2
Ij(u1)d[u1(x1)], (2.20)
where for j = 2, . . . , n, Ij(u1) is defined by
Ij(u1) =
∫
Dj
K(1, t2, . . . , tn)
n∏
i=2
ti
Ãi−ε/pidt2 . . . dtn,
and Dj =
{
(t2, t3, . . . , tn); 0 < tj ≤
1
u1(x1)
, 0 < tk <∞, k 6= j
}
.
Without losing generality, it is enough to estimate the integral I2(x1). Obviously, since 1−tε2 → 1
(t2 → 0+), there exists the constant C ≥ 0 such that 1 − tε2 ≤ C (t2 ∈ (0, 1]). Now, by using the
well-known Fubini’s theorem, it follows that
0 ≤ ε
∞∫
1
[u1(x1)]−1−εI2(u1)d[u1(x1)] =
= ε
∞∫
1
[u1(x1)]−1−ε
∫
(0,∞)n−2
1/u1(x1)∫
0
K(1, t2, . . . , tn)
n∏
i=2
t
Ãi−ε/pi
i dt2 . . . dtn
d[u1(x1)] =
= ε
∫
(0,∞)n−2
1∫
0
K(1, t2, . . . , tn)
n∏
i=2
t
Ãi−ε/pi
i
1/t2∫
1
t−1−ε
1 dt1
dt2 . . . dtn =
= ε
∫
(0,∞)n−2
1∫
0
K(1, t2, . . . , tn)
n∏
i=2
t
Ãi−ε/pi
i
(
1
ε
(1− tε2)
)
dt2 . . . dtn ≤
≤ C
∫
(0,∞)n−2
1∫
0
K(1, t2, . . . , tn)
n∏
i=2
t
Ãi−ε/pi
i dt2 . . . dtn ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
ON SOME MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES IN A DISCRETE CASE 811
≤ C
∫
(0,∞)n−1
K(1, t2, . . . , tn)
n∏
i=2
t
Ãi−ε/pi
i dt2 . . . dtn =
= Ck
(
Ã2 −
ε
p2
, . . . , Ãn −
ε
pn
)
<∞.
Further, regarding the above derived relation and inequality (2.20), we have that
Ĩ ≥ 1
ε
k
(
Ã2 −
ε
p2
, . . . , Ãn −
ε
pn
)
− o(1). (2.21)
Finally, by comparing the relations (2.18) and (2.21), we conclude that L∗ ≤ M when ε → 0+,
which is an obvious contradiction. Hence, it follows that the constant factor L∗ is the best possible
in (2.15).
Clearly, the constant factor L∗ is also the best possible in the inequality (2.16) since the equiva-
lence keeps the best possible constant.
Theorem 2.2 is proved.
3. Some applications. This section is dedicated to the applications of our general results, i.e.,
Theorems 2.1 and 2.2, to some particular choices of homogeneous kernels K : (0,∞)n → R, differ-
entiable functions ui : (0,∞) → R, i = 1, . . . , n, and real parameters Ãi, i = 1, . . . , n, defined in
the previous section.
Here, we shall be concerned with the homogeneous function
K1(x1, . . . , xn) =
1
(x1 + . . .+ xn)λ
, λ > 0.
Note that the kernel K1 is symmetric, strictly decreasing in each variable, and
k(β1 − 1, . . . , βn−1 − 1) =
∫
(0,∞)n−1
∏n−1
i=1
tβi−1
i(
1 +
∑n−1
i=1
ti
)λdt1 . . . dtn−1 =
=
Γ
(
λ−
∑n−1
i=1
βi
)∏n−1
i=1
Γ(βi)
Γ(λ)
, (3.1)
where we used the integral formula derived in the paper [4]. Now, in the above described setting, as
an immediate consequence of Theorems 2.1 and 2.2, we get the following result.
Corollary 3.1. Suppose the parameters qn, pi, Aij , i, j = 1, . . . , n, and the functions ui:
(0,∞) → R, i = 1, . . . , n, are defined as in statement of Theorem 2.1. If the parameters Aij ,
i, j = 1, . . . , n, fulfill the set of conditions (2.10), then the inequalities
∞∑
mn=1
. . .
∞∑
m1=1
∏n
i=1
a(i)
mi(∑n
i=1
ui(mi)
)λ ≤
≤ L1
n∏
i=1
( ∞∑
mi=1
[ui(mi)]
−1−piÃi [u′i(mi)]
1−pi
(
a(i)
mi
)pi)1/pi
(3.2)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
812 M. KRNIĆ, P. VUKOVIĆ
and ∞∑
mn=1
[un(mn)](1−qn)(−1−pnÃn) ×
×
∞∑
mn−1=1
. . .
∞∑
m1=1
∏n
i=1
a(i)
mi(∑n
i=1
ui(mi)
)λ
qn
1/qn
≤
≤ L1
n−1∏
i=1
( ∞∑
mi=1
[ui(mi)]
−1−piÃi [u′i(mi)]
1−pi
(
a(i)
mi
)pi)1/pi
, (3.3)
where L1 =
∏n
i=1
Γ(Ãi + 1)
Γ(λ)
, hold for all nonnegative real sequences
(
a
(i)
mi
)
mi∈N
and are equiva-
lent. Moreover, the constant factor L1 is the best possible in both inequalities (3.2) and (3.3).
Remark 3.1. Note that inequalities (3.2) and (3.3) involve the parameters Ãi, i = 1, 2, . . . , n,
since the parameters Aij , i, j = 1, 2, . . . , n, satisfy the set of conditions (2.10).
The following remark describes the connection between our Corollary 3.1 and Theorem 1.1 in
detail.
Remark 3.2. It is obvious that our Corollary 3.1 is the generalization of Theorem 1.1 from the
Introduction (see also [2]). Namely if we substitute the power functions ui(xi) = (xi + β)α and the
parameters Ãi =
λ
ri
− 1, i = 1, . . . , n, in Corollary 3.1 we get the inequalities from Theorem 1.1
with the best possible constant factor
α1−n
Γ(λ)
∏n
i=1
Γ
(
λ
ri
)
.
We conclude this paper with yet another consequence of Corollary 3.1, known from the literature.
Remark 3.3. Let
Aii =
(n− λ)(pi − 1)
p2
i
and Aij =
λ− n
pipj
, i, j = 1, 2, . . . , n, i 6= j, (3.4)
where pi, i = 1, 2, . . . , n, are conjugate exponents. These parameters are symmetric and
n∑
i=1
Aij =
n∑
j=1
Aij =
(n− λ)(pi − 1)
p2
i
+
n∑
j=1,j 6=i
λ− n
pipj
=
n− λ
pi
1−
n∑
j=1
1
pj
= 0.
Moreover, the above defined parameters satisfy the set of conditions (2.10), so the resulting relations
will include the best possible constant factors.
Now, for the above choice of parameters Aij defined by (3.4), and the functions ui(xi) = xi, the
inequalities (3.2) and (3.3) respectively read
∞∑
mn=1
. . .
∞∑
m1=1
∏n
i=1
a(i)
mi(∑n
i=1
mi
)λ ≤ L2
n∏
i=1
( ∞∑
mi=1
mi
n−1−λ
(
a(i)
mi
)pi)1/pi
(3.5)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
ON SOME MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES IN A DISCRETE CASE 813
and ∞∑
mn=1
m(1−qn)(pn−λ−1)
n
∞∑
mn−1=1
. . .
∞∑
m1=1
∏n
i=1
a(i)
mi(∑n
i=1
mi
)λ
qn
1/qn
≤
≤ L2
n−1∏
i=1
( ∞∑
mi=1
mi
n−1−λ
(
a(i)
mi
)pi)1/pi
, (3.6)
where L2 =
1
Γ(λ)
∏n
i=1
Γ
(
pi + λ− n
pi
)
. Note that the condition λ ≤ min1≤i≤n{pi} must be sat-
isfied, so that the function ui belongs to the set H(piAij), i, j = 1, 2, . . . , n (see the statement of
Theorem 2.1). Moreover, since we consider the Gamma function with positive argument, inequali-
ties (3.5) and (3.6) hold under condition n−min1≤i≤n{pi} ≤ λ ≤ min1≤i≤n{pi}.
Finally, let us mention that our inequality (3.5) is a discrete variant of the appropriate integral
result from paper [4].
1. Hardy G. H., Littlewood J. E., Pólya G. Inequalities. – 2 nd ed. – Cambridge: Cambridge Univ. Press, 1967.
2. Huang Q. On a multiple Hilbert’s inequality with parameters // J. Inequal.& Appl. – 2010. – 2010. – Art. ID 309319.
3. Mitrinović D. S., Pečarić J. E., Fink A. M. Classical and new inequalities in analysis. – Dordrecht etc.: Kluwer Acad.
Publ., 1993.
4. Rassias T. M., Yang B. On the way of weight coefficient and research for the Hilbert-type inequalities // Math. Inequal.
and Appl. – 2003. – 6, № 2. – P. 625 – 658.
5. Vuković P. Note on Hilbert-type inequalities // Turk. J. Math. – 2011. – 35. – P. 1 – 10.
6. Yang B., Brnetić I., Krnić M., Pečarić J. Generalization of Hilbert and Hardy – Hilbert integral inequalities // Math.
Inequal. and Appl. – 2005. – 8, № 2. – P. 259 – 272.
Received 08.07.11
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
|
| id | umjimathkievua-article-2465 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:23:56Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c5/92766a27a3f3101e427c0ad9978a16c5.pdf |
| spelling | umjimathkievua-article-24652020-03-18T19:16:10Z On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case Про деякі багатовимірні нерівності гільбертового типу у дискретному випадку Krnić, M. Vuković, P. Крнич, М. Вукович, П. Motivated by the results of Huang, we deduce a pair of discrete multidimensional Hilbert-type inequalities involving a homogeneous kernel of negative degree. We also establish conditions under which the constant factors involved in the established inequalities are the best possible. Finally, we consider some particular settings with homogeneous kernels and weight functions. In this way, we obtain generalizations of some results known from the literature. З метою узагальнення результат Хуанга отримано дві дискретні багатовимiрнi нєрівності гільбертового типу з однорідним ядром від'ємного степеня. Також встановлено умови, за яких сталі множники, що входять до отриманих нерівностей, є найкращими з можливих. Розглянуто деякі конкретні випадки однорідних ядер та вагових функцій. Це дає змогу узагальнити деякі відомі результати. Institute of Mathematics, NAS of Ukraine 2013-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2465 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 6 (2013); 802–813 Український математичний журнал; Том 65 № 6 (2013); 802–813 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2465/1696 https://umj.imath.kiev.ua/index.php/umj/article/view/2465/1697 Copyright (c) 2013 Krnić M.; Vuković P. |
| spellingShingle | Krnić, M. Vuković, P. Крнич, М. Вукович, П. On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case |
| title | On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case |
| title_alt | Про деякі багатовимірні нерівності гільбертового типу у дискретному випадку |
| title_full | On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case |
| title_fullStr | On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case |
| title_full_unstemmed | On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case |
| title_short | On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case |
| title_sort | on some multidimensional hilbert-type inequalities in the discrete case |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2465 |
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