INVERSES OF NEW HILBERT-PACHPATTE-TYPE
INEQUALITIES
CHANG-JIAN ZHAO AND WING-SUM CHEUNG
Received 7 February 2006; Revised 3 June 2006; Accepted 5 June 2006
We establish a new inequality of Hilbert type for a finite double number of nonnegative
sequences of real numbers and some interrelated results, which are inverse and general
forms of Pachpatte’s and Handley’s results. An integral version and some interrelated
results are also obtained. These results provide some new estimates on such types of inequalities.
Copyright © 2006 C.-J. Zhao and W.-S. Cheung. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The various generalizations and sharpenings of Hilbert’s double series inequality and its
integral version are obtained by Pachpatte, Handley et al., Gao et al. (see [1, 3, 5, 7, 10,
12, 16–18]). An elegant survey on this kind of inequalities was provided by Yang and
Rassias (see [13]). Moreover, Pachpatte [9] established a new Hilbert-type inequality and
its integral version as follows.
Theorem 1.1. Let {am }, {bn } be two nonnegative sequences of real numbers defined for m =
1,2,...,k and n = 1,2,...,r with a0 = b0 = 0, and let { pm }, {qn } be two positive sequences
1,2,...,k, n = 1,2,...,r, where k, r are natural numbers.
of real numbers
defined for m =
m
n
Define Pm = s=1 ps and Qn = t=1 qt . Let φ and ψ be two real-valued nonnegative, convex,
and submultiplicative functions defined on R+ = [0, ∞). Then
r
k φ am ψ bn
m=1 n=1
m+n
≤ M(k,r)
k
(k − m + 1) pm φ
m=1
×
r
(r − n + 1) qn ψ
n =1
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 97860, Pages 1–11
DOI 10.1155/JIA/2006/97860
∇bn
qn
∇ am
2 1/2
pm
(1.1)
2 1/2
,
2
Inverses of new Hilbert-Pachpatte-type inequalities
where
1
M(k,r) =
2
k
m=1
φ Pm
Pm
2 1/2 r
n=1
φ Qn
Qn
2 1/2
,
(1.2)
and ∇am = am − am−1 , ∇bn = bn − bn−1 .
Theorem 1.2. Let f ∈ C 1 [[0,x), R+ ], g ∈ C 1 [[0, y), R+ ] with f (0) = g(0) = 0 and let
p(σ), q(τ) be two positive functions defined for σ ∈ [0,x) and τ ∈ [0, y). Let P(s) =
s
t
0 p(σ)dσ and Q(t) = 0 q(τ)dτ for s ∈ [0,x) and t ∈ [0, y), where x, y are positive real
numbers. Let φ, and ψ be as in Theorem 1.1. Then
x y 0
0
φ f (s) ψ g(t)
dsdt ≤ L(x, y)
s+t
×
x
0
(x − s) p(s)φ
g (t)
(y − t) q(y)φ
q(t)
0
y
2
f (s)
p(s)
1/2
ds
2
(1.3)
1/2
dt
,
where
1
L(x, y) =
2
x
φ P(s)
P(s)
0
2
1/2 y
ds
φ Q(t)
Q(t)
0
2
1/2
dt
,
(1.4)
and denotes the derivative of a function.
In [4], Handley et al. gave general versions of inequalities (1.1) and (1.3) as follows.
Theorem 1.3. Let {ai,mi } (i = 1,2,...,n) be n sequences of nonnegative real numbers defined
for mi = 1,2,...,ki with a1,0 = a2,0 = · · · = an,0 = 0, and let { pi,mi } be n sequences of positive real numbers
defined for mi = 1,2,...,ki , where ki (i = 1,2,...,n) are natural numbers.
i
p
Set Pi,mi = m
si =1 i,si . Let φi (i = 1,2,...,n) be n real-valued nonnegative convex and submultiplicative
functions
defined on R+ . Let αi ∈ (0,1), and set αi = 1 − αi (i = 1,2,...,n),
n
n
α = i=1 αi , and α = i=1 αi = n − α. Then
k1
kn
n
i=1 φi ai,mi
···
n α
m1 =1
mn =1
i=1 αi mi
≤ M k1 ,...,kn
n
i =1
ki
ki − mi + 1
pi,mi φi
mi =1
∇ai,mi
(1.5)
1/αi αi
,
pi,mi
where
M k1 ,...,kn
n
1 = α
(α ) i=1
and ∇ai,mi = ai,mi − ai,mi −1 (i = 1,2,...,n).
ki
mi =1
φi Pi,mi
Pi,mi
1/αi αi
,
(1.6)
C.-J. Zhao and W.-S. Cheung 3
Theorem 1.4. Let fi ∈ C 1 [[0,ki ], R+ ], i = 1,2,...,n, with fi (0) = 0 (i =1,2,...,n), let
s
pi (σi ) be n positive functions defined for σi ∈ [0,xi ] (i = 1,...,n). Set Pi (si ) = 0 i pi (σi )dσi for
si ∈ [0,xi ], where xi are positive real numbers. Let φi , αi , αi , α, and α be as in Theorem 1.3.
Then
x1
0
···
x n n
0
i=1 φi fi si
n α
i=1 αi si
≤ L x1 ,...,xn
n
ds1 · · · dsn
0
i=1
1/αi αi
f si
xi − si pi si φi i dsi ,
xi (1.7)
pi si
where
L x1 ,...,xn
αi
n xi 1/αi
φi Pi si
1 dsi .
= α
(α ) i=1 0
Pi si
(1.8)
The main purpose of the present paper is to establish their reverse versions, which
are more extensive results for this type of inequalities. Our main results are given in the
following theorems.
Theorem 1.5. Let {asi ,ti ,msi ,mti } (i = 1,2,...,n) be n sequences of nonnegative numbers defined for msi = 1,2,...,ksi , mti = 1,2,...,kti , with asi ,ti ,0,mti = asi ,ti ,msi ,0 = 0 (i = 1,2,...,n),
where ksi and kti (i = 1,2,...,n) are natural numbers. Let { psi ,ti ,msi ,mti } be n sequences of
positive real numbers defined for msi , mti . Set
Psi ,ti ,msi ,mti =
msi
mti
mξi =1 mηi =1
psi ,ti ,mξi ,mηi
(i = 1,2,...,n).
(1.9)
funcLet φi (i = 1,2,...,n) be n real-valued nonnegative concave and supermultiplicative
n
tions
defined on R+ . Let αi ∈ (1, ∞). Set αi = 1 − αi (i = 1,2,...,n), α = i=1 αi , and α =
n
i=1 αi = n − α. Define operators
∇1 asi ,ti ,msi ,mti = asi ,ti ,msi ,mti − asi ,ti ,msi −1,mti ,
∇2 asi ,ti ,msi ,mti = asi ,ti ,msi ,mti − asi ,ti ,msi ,mti −1 ,
∇2 ∇1 asi ,ti ,msi ,mti = ∇2 ∇1 asi ,ti ,msi ,mti .
(1.10)
Then
ks1
kt1
···
ms1 =1 mt =1
ksn
ktn
msn =1 mtn =1
n
i=1 φi
(1/α )
n
asi ,ti ,msi ,mti
i=1 αi msi mti
α
≥ C ks1 kt1 ,...,ksn ktn
×
ksi
n i=1
kti
msi =1 mti =1
ksi − msi +1 kti − mti +1
psi ,ti ,msi ,mti φi
∇2 ∇1 asi ,ti ,msi ,mti
psi ,ti ,msi ,mti
1/αi αi
,
(1.11)
4
Inverses of new Hilbert-Pachpatte-type inequalities
where
C ks1 kt1 ,...,ksn ktn =
n
ksi
kti
msi =1 mti =1
i =1
φi Psi ,ti ,msi ,mti
Psi ,ti ,msi ,mti
1/αi αi
.
(1.12)
Theorem 1.6. Let fi (si ,ti ) (i = 1,2,...,n) be real-valued continuous functions defined on
[0,xi ) × [0, yi ), where xi ∈ (0, ∞), yi ∈ (0, ∞), and with fi (0,ti ) = fi (si ,0) = 0 (i = 1,2,... ,n).
Let pi (σi ) and qi (τi ) be positive continuous functions defined for σi ∈ (0,si ), τi ∈ (0,ti ). Set
Pi si ,ti =
ti s i
0
0
pi σi qi τi dσi dτi .
(1.13)
For the function fi (si ,ti ), denote the partial derivatives (∂/∂si ) fi (si ,ti ), (∂/∂ti ) fi (si ,ti ), and
(∂2 /∂si ∂ti ) fi (si ,ti ) by D1 fi (si ,ti ), D2 fi (si ,ti ), and D2 D1 fi (si ,ti ) = D1 D2 fi (si ,ti ), respectively.
Let φi , αi , αi , α, and α be as in Theorem 1.5. Then
x1 y 1
0
0
···
xn y n n
0
0
φi fi si ,ti
(1/α )
i =1
n
i=1 αi si ti
α ds1 dt1 · · · dsn dtn
= G x1 y1 ,...,xn yn
×
n xi y i
0
i=1
0
xi − si yi − ti
pi si qi ti · φi
αi
1/αi
D2 D1 fi si ,ti
pi si qi ti
dsi dti
,
(1.14)
where
G x1 y1 ,...,xn yn
αi
1/αi
n xi y i φi Pi si ,ti
dsi dti .
=
i=1
0
Pi si ,ti
0
(1.15)
These results provide new estimates on these types of inequalities.
2. Proofs and remarks
Proof of Theorem 1.5. From the operators defined in Theorem 1.5, we obtain
asi ,ti ,msi ,mti =
=
msi
mξi =1
msi
asi ,ti ,mξi ,mti − asi ,ti ,mξi −1,mti
mti
mξi =1 mηi =1
=
msi
mti
mξi =1 mηi =1
asi ,ti ,mξi ,mηi − asi ,ti ,mξi −1,mηi − asi ,ti ,mξi ,mηi −1 + asi ,ti ,mξi −1,mηi −1
∇2 ∇1 asi ,ti ,mξi ,mηi .
(2.1)
C.-J. Zhao and W.-S. Cheung 5
From (2.1) and the hypotheses of Theorem 1.5, and in view of Jensen’s inequality and
inverse Hölder’s inequality [6], we obtain
φi asi ,ti ,msi ,mti
= φi
Psi ,ti ,msi ,mti
msi
mξi =1
mηi =1
msi
mti
mξi =1 mηi =1
psi ,ti ,mξi ,mηi φi
φi Psi ,ti ,msi ,mti α
ms i mt i i ×
≥
Psi ,ti ,msi ,mti
psi ,ti ,mξi ,mηi
psi ,ti ,mξi ,mηi ∇2 ∇1 asi ,ti ,mξi ,mηi psi ,ti ,mξi ,mηi
mξi =1
msi
mti
mξi =1
≥ φi Psi ,ti ,msi ,mti · φi
mηi =1 psi ,ti ,mξi ,mηi ∇2 ∇1 asi ,ti ,mξi ,mηi
msi mti
mξi =1
mηi =1 psi ,ti ,mξi ,mηi
msi
φi Psi ,ti ,msi ,mti
≥
Psi ,ti ,msi ,mti
mti
msi
mti
mηi =1
psi ,ti ,mξi ,mηi
∇2 ∇1 asi ,ti ,mξi ,mηi
psi ,ti ,mξi ,mηi
mti
psi ,ti ,mξi ,mηi φi
mξi =1 mηi =1
∇2 ∇1 asi ,ti ,mξi ,mηi
1/αi αi
.
psi ,ti ,mξi ,mηi
(2.2)
On the other hand, noticing αi < 0 (i = 1,2,...,n), λi > 0 (i = 1,2,...,n), and applying the
well-known means inequality, we have
n
1 ≥
α λi
α i=1 i
α
λi i
i =1
n
α
.
(2.3)
Hence
n
i=1 φi
(1/α )
≥
n
asi ,ti ,msi ,mti
i=1 αi msi mti
α
msi
n
φi Psi ,ti ,msi ,mti
i=1
Psi ,ti ,msi ,mti
mti
mξi =1 mηi =1
psi ,ti ,mξi ,mηi φi
∇2 ∇1 asi ,ti ,mξi ,mηi
(2.4)
1/αi αi
psi ,ti ,mξi ,mηi
.
Taking the sum of both sides of (2.4) over msi , mti (i = 1,2,...,n) from 1 to ksi , ksi , respectively, and in view of inverse Hölder’s inequality, we get
ks1
kt1
···
ms1 =1 mt =1
ksi
n ≥
ktn
ksn
msn =1 mtn =1
kti
i=1 msi =1 mti =1
n
i=1 φi
(1/α )
n
φi Psi ,ti ,msi ,mti
Psi ,ti ,msi ,mti
asi ,ti ,msi ,mti
i=1 αi msi mti
msi
α
mti
mξi =1 mηi =1
psi ,ti ,mξi ,mηi φi
∇2 ∇1 asi ,ti ,mξi ,mηi
psi ,ti ,mξi ,mηi
1/αi αi
6
Inverses of new Hilbert-Pachpatte-type inequalities
≥
n
ksi
kti
φi Psi ,ti ,msi ,mti
Psi ,ti ,msi ,mti
msi =1 mti =1
i=1
ksi
×
kti
msi =1 mti =1
mti
msi
mξi =1 mηi =1
1/αi αi
psi ,ti ,mξi ,mηi φi
∇2 ∇1 asi ,ti ,mξi ,mηi
1/αi αi
psi ,ti ,mξi ,mηi
= C ks1 kt1 ,...,ksn ktn
×
ksi
n kti
ksi − mξi +1 kti − mηi +1
i=1 mξi =1 mηi =1
psi ,ti ,mξi ,mηi φi
∇2 ∇1 asi ,ti ,mξi ,mηi
1/αi αi
psi ,ti ,mξi ,mηi
= C ks1 kt1 ,...,ksn ktn
×
ksi
n kti
i=1 msi =1 mti =1
ksi − msi +1 kti − mti +1 psi ,ti ,msi ,mti φi
∇2 ∇1 asi ,ti ,msi ,mti
1/αi αi
.
psi ,ti ,msi ,mti
(2.5)
This completes the proof.
Remark 2.1. Let asi ,ti ,msi ,mti , msi , mti , ksi , kti , asi ,ti ,0,mti , asi ,ti ,msi ,0 , Psi ,ti ,msi ,mti , ∇1 , and ∇2 be as
in Theorem 1.5. Let φi , αi , αi , α, and α (i = 1,2,...,n) be as in Theorem 1.3. Similar to
the proof of Theorem 1.5, we obtain
ks1
kt1
···
ms1 =1 mt =1
ksn
ktn
n
i=1 φi asi ,ti ,msi ,mti
n α
msn =1 mtn =1
i=1 αi msi mti
≤ C ks1 kt1 ,...,ksn ktn
×
n
ksi
kti
msi =1 mti =1
i=1
ksi − msi +1 kti − mti +1 psi ,ti ,msi ,mti φi
∇2 ∇1 asi ,ti ,msi ,mti
1/αi αi
,
psi ,ti ,msi ,mti
(2.6)
where
C ks1 kt1 ,...,ksn ktn
n
1 = α
(α ) i=1
ksi
kti
msi =1 mti =1
φi Psi ,ti ,msi ,mti
Psi ,ti ,msi ,mti
1/αi αi
.
(2.7)
This is just a general form of inequality (1.5) and the inverse form of inequality (1.11).
C.-J. Zhao and W.-S. Cheung 7
Moreover, let {ai,mi }, mi , ki , ai,0 , pi,mi , ∇, and Pi,mi (i = 1,2,...,n) be as in Theorem 1.3.
Let φi , αi , αi , α, and α (i = 1,2,...,n) be as in Theorem 1.5. Similar to the proof of
Theorem 1.5, we obtain
k1
n
kn
i=1 φi ai,mi
···
n α
m1 =1
mn =1 (1/α ) i=1 αi mi
≥ M k1 ,...,kn
n
ki
ki − mi + 1
pi,mi φi
∇ai,mi
,
pi,mi
mi =1
i =1
(2.8)
1/αi αi
where
n
M k1 ,... ,kn =
ki
mi =1
i =1
φi Pi,mi
Pi,mi
1/αi αi
.
(2.9)
This is just an inverse form of inequality (1.5) which was given by Handley et al. [4].
Let n = 2, α1 = α2 = 2, then α1 = α2 = −1, putting them in inequality (2.8), we have
k2
k1
φ1 a1,m1 φ2 a2,m2
−2
m 1 + m2
m1 =1 m2 =1
≥ M k1 ,k2
×
k1
k1 − m1 + 1
m1 =1
k2
k2 − m2 + 1
p1,mi φ1
p2,m2 φ2
1/2 2
(2.10)
p1,m1
∇a2,m2
m2 =1
∇a1,m1
1/2 2
p2,m2
,
where
M k1 ,k2 = 4
k1
m1 =1
φ1 P1,m1
P1,m1
−1 −1 k2
m2 =1
φ2 P2,m2
P2,m2
−1 −1
.
(2.11)
This is just an inverse form of inequality (1.1) which was given by Pachpatte [9].
Proof of Theorem 1.6. From the hypotheses of Theorem 1.6, we obtain
fi si ,ti =
s i ti
0
0
D2 D1 fi σi ,τi dσi dτi .
(2.12)
8
Inverses of new Hilbert-Pachpatte-type inequalities
From (2.12) and using Jensen’s integral inequality and Hölder’s integral inequality [6],
we obtain
φi fi si ,ti
= φi
Pi si ,ti
≥ φi Pi si ,ti
ti s i
ti s i
0 0
· φi
t s
i
i
φi (Pi si ,ti
Pi si ,ti
≥
0
pi σi qi τi D2 D1 fi σi ,τi pi σi qi τi dσi dτi
ti s i 0 0 pi σi qi τi dσi dτi
0 0
0
φi Pi si ,ti αi
≥
si ti
Pi si ,ti
pi σi qi τi D2 D1 fi σi ,τi / pi σi qi τi dσi dτi
ti s i 0 0 pi σi qi τi dσi dτi
pi σi qi τi · φi
ti s i
0
0
D2 D1 fi σi ,τi
pi σi qi τi
pi σi qi τi · φi
dσi dτi
D2 D1 fi σi ,τi
pi σi qi τi
αi
1/αi
dσi dτi
.
(2.13)
From the well-known inequality for means, we have
n
φi fi si ,ti
(1/α )
i=1
n
i=1 αi si ti
α
αi
t s 1/αi
n
i
i
φi Pi si ,ti
D2 D1 fi σi ,τi
≥
pi σi qi τi · φi
dσi dτi .
Pi si ,ti
i=1
0
pi σi qi τi
0
(2.14)
Integrating both sides of (2.14) over si , ti from 1 to xi , yi (i = 1,2,...,n) and in view of
Hölder’s integral inequality and Fubini’s theorem, we observe that
x1 y 1
0
0
≥
···
xn y n n
0
0
φi fi si ,ti
(1/α )
i=1
n
i=1 αi si ti
α ds1 dt1 · · · dsn dtn
n xi y i
φi Pi si ,ti
i=1 0
Pi si ,ti
0
ti s i
×
0
0
pi σi qi τi · φi
1/αi
D2 D1 fi σi ,τi
pi σi qi τi
αi
dσi dτi
dsi dti
αi
1/αi
n xi y i φi Pi si ,ti
≥
dsi dti
i=1
×
0
Pi si ,ti
0
xi y i
ti s i
0
0
0
0
pi σi qi τi · φi
1/αi
D2 D1 fi σi ,τi
pi σi qi τi
dσi dτi dsi dti
αi
C.-J. Zhao and W.-S. Cheung 9
= G x1 y1 ,...,xn yn
×
n xi y i
0
i=1
xi − si yi − ti
0
pi si qi ti · φi
αi
1/αi
D2 D1 fi si ,ti
pi si qi ti
dsi dti
.
(2.15)
The proof is complete.
Remark 2.2. Let fi (si ,ti ), xi , yi , fi (0,ti ), fi (si ,0), D1 fi (si ,ti ), D2 fi (si ,ti ), D2 D1 fi (si ,ti ), pi (σi ),
qi (τi ), and Pi (si ,ti ) be as in Theorem 1.6. Let φi , αi , αi , α, and α be as in Theorem 1.3.
Similar to the proof of Theorem 1.6, we have
x1 y 1
0
0
···
xn y n n
0
0
φi fi si ,ti
α ds1 dt1 · · · dsn dtn
n
i=1 αi si ti
i =1
≤ G x1 y1 ,...,xn yn
×
n xi y i
0
i=1
xi − si yi − ti
0
pi si qi ti · φi
1/αi
D2 D1 fi si ,ti
pi si qi ti
αi
dsi dti
,
(2.16)
where
G x1 y1 ,...,xn yn
αi
1/αi
n xi y i φi Pi si ,ti
1 dsi dti .
= α
(α ) i=1 0 0
Pi si ,ti
(2.17)
This is just a general form of inequality (1.7) and an inverse form of the inequality
(1.14).
On the other hand, let fi , fi (0), fi (si ), pi (σi ), σi , ki , si , xi , Pi (si ) be as in Theorem 1.4,
and let φi , αi , αi , α, α be as in Theorem 1.5. Then
x1
0
···
xn
0
n
i=1 φi
(1/α )
n
≥ L x1 ,...,xn
fi si
i=1 αi si
n
1/αi αi
f si
xi − si pi si φi i dsi ,
xi 0
i =1
α ds1 · · · dsn
(2.18)
pi si
where
L x1 ,...,xn
αi
n xi 1/αi
φi Pi si
dsi .
=
i=1
0
Pi si
(2.19)
This is just an inverse form of inequality (1.7) which was given by Handley et al. [4].
10
Inverses of new Hilbert-Pachpatte-type inequalities
Let n = 2, α1 = α2 = 2, then α1 = α2 = −1, taking them in inequality (2.18), we have
x1 x2
0
φ1 f1 s1 φ2 f2 s2
s1 + s2
0
≥ L x1 ,x2
×
ds1 ds2
x1 0
x2 0
−2
x2 − s2
x1 − s1
p1 s1 φ1
p2 s2 φ2
1/2
f2 s1
p1 s1
ds1
(2.20)
2
1/2
f2 s2
p2 s2
2
ds2
,
where
L x1 ,x2 = 4
x1
0
−1 −1
φ1 P1 s1
P1 s1
ds1
x2
0
−1
−1
φ2 P2 s2
P2 s2
ds2
.
(2.21)
This is just an inverse form of inequality (1.3) which was given by Pachpatte [9].
For interrelated research on similar inequalities, one is referred to [2, 8, 11, 14, 15],
and the references cited therein.
Acknowledgments
Research of the first author is supported by Zhejiang Provincial Natural Science Foundation of China (Y605065), National Natural Sciences Foundation of China (10271071),
Foundation of the Education Department of Zhejiang Province of China (20050392),
and the Academic Mainstay of Middle-age and Youth Foundation of Shandong Province
of China (200203). Research of the second author is partially supported by the Research
Grants Council of the Hong Kong SAR, China (Project no. HKU7040/03P, and HKU
7017/05P).
References
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Chang-Jian Zhao: Department of Information and Mathematics Sciences, College of Science,
China Jiliang University, Hangzhou 310018, China
E-mail address: chjzhao@cjlu.edu.cn
Wing-Sum Cheung: Department of Mathematics, The University of Hong Kong, Pokfulam Road,
Hong Kong
E-mail address: wscheung@hku.hk