Volume 2010, Article ID 486127,11pages doi:10.1155/2010/486127
Research Article
A Hilbert Integral-Type Inequality with Parameters
Shang Xiaozhou and Gao Mingzhe
Department of Mathematics and Computer Science, Normal College of Jishou University, Hunan Jishou 416000, China
Correspondence should be addressed to Gao Mingzhe,[email protected]
Received 15 March 2010; Revised 9 April 2010; Accepted 14 April 2010
Academic Editor: Feng Qi
Copyrightq2010 S. Xiaozhou and G. Mingzhe. 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.
A Hilbert-type integral inequality with parameters α and α, λ > 0 can be established by introducing a nonhomogeneous kernel function. And the constant factor is proved to be the best possible. And then some important and especial results are enumerated. As applications, some equivalent forms are studied.
1. Introduction
Letfx, gx∈L20,∞. Then
∞
0
fxgy
xy dx dy≤π
∞
0
f2xdx
1/2∞
0
g2xdx
1/2
, 1.1
where the constant factorπ is the best possible, and the equality in1.1holds if and only iffx 0, orgx 0.This is the famous Hilbert integral inequalitysee1,2. Owing to the importance of the Hilbert inequality in analysis and applications, some mathematicians have been studying them. Recently, various improvements and extensions of 1.1 appear in a great deal of paperssee3–11, etc.. Specially, Gao and Hsu enumerated the research articles to more than 40 in the paper 6. It is obvious that the integral kernel function of the left hand side of1.1is a homogeneous form of degree−1. In general, the Hilbert type integral inequality with a homogeneous kernel of degree−1 has been studied in the paper
For convenience, we need to introduce the Catalan constant and define a real function. The Catalan constant is defined by
G 1
2 1
0
Kdk
∞
n0
−1n
2n12, 1.2
whereKis the complete elliptic integral, namely,
KKk
π/2
0
dθ
√
1−k2sin2θ. 1.3
And the approximation ofGis that
G0,915965594· · · . 1.4
These results can be found in the paper12, page 503. Letλ >0. Define a real function by
Sλ
∞
n0
−1n 1
2n1λ1. 1.5
In particular,S1 G, whereGis the Catalan constant.
In order to prove our main results, we need the following lemmas.
Lemma 1.1. Letabe a positive number andb >−1. Then
∞
0
xbe−axdx Γb1
ab1 . 1.6
Proof. According to the definition of the gamma function, we obtain immediately1.6. This result can be also found in the paper12, page 226, formula1053.
Lemma 1.2. Letα, λ >0. Define a function by
Cα, λ 2
2
α
λ1
Γλ1Sλ, 1.7
whereΓzis the gamma function, andSλis a function defined by1.5. Then ∞
0
k1, uuα/2−1duCα, λ, 1.8
Proof. It is easy to deduce that
∞
0
k1, uuα/2−1du
1
0
ln1/uλ
1uα u
α/2−1du ∞
1
lnuλ
1uαu α/2−1du
1
0
ln1/uλ
1uα u
α/2−1du 1
0
ln1/vλ
1vα u
α/2−1du
2 1
0
ln1/uλ
1uα u
α/2−1du2 ∞
0
tλe−α/2t 1e−αt dt.
1.9
Expanding 1/1e−αtinto power series ofe−αtand then using Lemma1.1, we have
∞
0
tλe−α/2t 1e−αt dt
∞
0
tλe−α/2t1−e−αte−2αte−3αt−e−4αt· · ·dt
∞
0
tλe−α/2t−tλe−3/2αttλe−5/2αt−tλe−7/2αt· · ·dt
2
α
λ1
Γλ1
∞
n0
−1n 1
2n1λ1
2
α
λ1
Γλ1Sλ.
1.10
It follows from1.9,1.10, and1.7that the equality1.8holds.
2. Main Results
In the section we will formulate our main results.
Theorem 2.1. Letfandgbe two real functions, and letαandλbe arbitrary two positive numbers. If ∞
0 x1−αf2xdx <∞and ∞
0 x1−αg2xdx <∞, then
∞
0
lnxyλ
fxgy
1xyα dx dy≤Cα, λ
∞
0
x1−αf2xdx
1/2∞
0
x1−αg2xdx 1/2
, 2.1
Proof. We may apply Hardy’s technique and Cauchy-Schwarz’s inequality to estimate the left-hand side of2.1as follows:
∞
0
lnxyλ
fxgy
1 xyα dx dy
∞
0
lnxyλ/2
fx
1xyα1/2
x y
2−α/4 lnxyλ/2fx
1 xyα1/2
y
x
2−α/4
gydx dy
≤
∞
0
lnxyλ 1 xyα
x y
2−α/2
f2xdx dy
1/2∞
0
lnxyλ 1xyα
y
x
2−α/2
g2ydx dy
1/2
∞
0
ωα, λ, xf2xdx
1/2∞
0
ωα, λ, xg2xdx
1/2
,
2.2
whereωα, λ, x 0∞|lnxy|λ/1 xyαx/y2−α/2dy. By substitutinguforxy, it is easy to deduce that
ωα, λ, x
∞
0
lnxyλ
1xyα
x y
2−α/2
dy
x1−α
∞
0 |lnu|λ
1uα
1
u
2−α/2
du
x1−α
∞
0
|ln1/u|λ
1uα u
α/2−1du.
2.3
Based on1.8, we obtain
ωα, λ, x x1−αCα, λ. 2.4
It is known from2.2and2.4that the inequality2.1is valid.
Iffx 0, orgx 0, then the equality in2.1obviously holds. Iffx/0 and
gx/0, then
0<
∞
0
x1−αf2xdx <∞, 0< ∞
0
If2.2takes the form of the equality, then there exists a pair of non-zero constantsc1 andc2such that
c1
lnxyλ
1xyαf
2x
x y
1−α/2
c2
lnxyλ
1xyαg
2yy
x
1−α/2
a.e.on0,∞×0,∞.
2.6
Then we have
c1x2−αf2x c2y2−αg2
yC0.constant a.e.on0,∞×0,∞. 2.7
Without losing the generality, we suppose thatc1/0; then
∞
0
x1−αf2xdx C0 c1
∞
0
x−1dx. 2.8
This contradicts that 0<0∞x1−αf2xdx <∞. Hence it is impossible to take the equality in
2.2. It shows that it is also impossible to take the equality in2.1.
It remains to need only to show thatCα, λin2.1is the best possible. Below we will apply Yang’s techniquesee13to verify our assertion.
For alln∈N, define two functions by
fnx ⎧ ⎨ ⎩
xα/2−11/2n x∈0,1,
0 x∈1,∞,
gnx ⎧ ⎨ ⎩
0 x∈0,1, xα/2−1−1/2n x∈1,∞.
2.9
Then we have
1
0
x1−αfn2xdx
1/2
∞
1
x1−αg2nxdx
1/2
√n. 2.10
Let 0≤k≤Cα, λsuch that the inequality2.1is still valid when Cα, λis replaced byk. Specially, we have
1
n
∞
0
lnxyλ
fnxgn
y
1xyα dx dy≤k
1
n
∞
0
x1−αfn2xdx
1/2∞
0
x1−αgn2xdx
1/2
k.
Letk1, xy |lnxy|λ/1 xyα. By Fubini’s theorem, we have
k≥ 1 n
∞
0
k1, xyfnxgn
ydx dy
1
n
∞
1
yα/2−1−1/2n
1
0
k1, xyxα/2−11/2ndx
dy
1
n
∞
1
y−1−1/n y
0
k1, uuα/2−11/2ndu dy
1
n
∞
1
y−1−1/n
1
0
k1, uuα/2−11/2ndu
dy
∞
1
y−1−1/n
y
1
k1, uuα/2−11/2ndu dy
1
n
∞
1
n
1
0
k1, uuα/2−11/2ndu
∞
1
k1, uuα/2−11/2n
∞
u
y−1−1/ndy du
1
0
k1, uuα/2−11/2ndu
∞
1
k1, uuα/2−1−1/2ndu.
2.12
By Fatou’s lemma and1.8, we have
k≥ lim n→ ∞
1
0
k1, uuα/2−11/2ndu lim n→ ∞
∞
1
k1, uuα/2−1−1/2ndu
≥ 1
0 lim
n→ ∞k1, uu
α/2−11/2ndu ∞
1 lim
n→ ∞k1, uu
α/2−1−1/2ndu
1
0
k1, uuα/2−1du
∞
1
k1, uuα/2−1du
∞
0
k1, uuα/2−1duCα, λ.
2.13
Based on Theorem2.1, we may build some important and interesting inequalities.
Theorem 2.2. Let n be a nonnegative integer and α > 0. If 0∞x1−αf2xdx < ∞ and ∞
0 x1−αg2xdx <∞, then
∞
0
lnxy2nfxgy
1xyα dx dy≤Cα,2n
∞
0
x1−αf2xdx
1/2∞
0
x1−αg2xdx
1/2
,
2.14
whereCα,2n π/α2n1En,theEnsare the Euler numbers, namely,E01, E11, E25, E3
61, E4 1385,and so forth, and the constant factorCα,2nin2.14is the best possible. And the equality in2.14holds if and only iffx 0, orgx 0.
Proof. We need only to verify the constant factorCα,2nin2.14. Whenλ2n, it is known from1.10that
Cα,2n 2
2
α
2n1
Γ2n1
∞
k0
−1k 1
2k12n1. 2.15
It is known from the paper14that
∞
k0
−1k 1
2k12n1
π2n1
22n22n!En, 2.16
where theEnsare the Euler numbers, namely,E0 1, E1 1, E2 5, E3 61, E4 1385,and so forth.
It follows that the constant factorCα,2nin2.14is correct.
In particular, based on Theorem2.2, the following results are gotten.
Corollary 2.3. If0∞f2xdx <∞and∞
0 g2xdx <∞, then
∞
0
fxgy
1xy dx dy≤π
∞
0
f2xdx
1/2∞
0
g2xdx 1/2
, 2.17
where the constant factorπin2.17is the best possible. And the equality in2.17holds if and only iffx 0, orgx 0.
Corollary 2.4. If0∞f2xdx <∞and∞
0 g2xdx <∞, then
∞
0
lnxy2fxgy
1xy dx dy≤π
3
∞
0
f2xdx
1/2∞
0
g2xdx
1/2
, 2.18
Theorem 2.5. Letfandgbe two real functions, andαa positive number. If0∞x1−αf2xdx <∞ and0∞x1−αg2xdx <∞, then
∞
0
lnxyfxg
y
1xyα dx dy≤
8G α2
∞
0
x1−αf2xdx
1/2∞
0
x1−αg2xdx
1/2
, 2.19
whereGis the Catalan constant, that is,G 0,915965594· · ·. And the constant factor 8G/α2in
2.19is the best possible. And the equality in2.19holds if and only iffx 0, orgx 0.
Proof. We need only to verify the constant factor in2.19. For caseλ1, based on1.7,1.5
and1.2, it is easy to deduce that the constant factor is thatCα,1 8G/α2.
In particular, whenαλ1, the following result is obtained
Corollary 2.6. Letfandgbe two real functions. If0∞f2xdx <∞and∞
0 g2xdx <∞, then
∞
0
lnxyfxg
y
1xy dx dy≤8G
∞
0
f2xdx
1/2∞
0
g2xdx
1/2
, 2.20
whereGis the Catalan constant, that is,G0,915965594· · ·. And the constant factor 8Gin2.20 is the best possible. And the equality in2.20holds if and only iffx 0, orgx 0.
A great deal of the Hilbert type inequalities can be established provided that the parameters are properly selected
3. Some Equivalent Forms
As applications, we will build some new inequalities.
Theorem 3.1. Let f be a real function, and let α and λ be arbitrary two positive numbers. If ∞
0 x1−αf2xdx <∞, then
∞
0
yα−1
∞
0
lnxyλ
1xyαfxdx
2
dy≤Cα, λ2
∞
0
x1−αf2xdx, 3.1
Proof. First, we assume that the inequality2.1is valid. Define a functiongyby
gyyα−1 ∞
0
lnxyλ
1xyαfxdx, y∈0,∞. 3.2
By using2.1, we have
∞
0
yα−1
∞
0
lnxyλ
1xyαfxdx
2
dy
∞
0
lnxyλ 1xyαfxg
ydx dy
≤Cα, λ
∞
0
x1−αf2xdx
1/2∞
0
y1−αg2ydy 1/2
Cα, λ
∞
0
x1−αf2xdx
1/2⎧⎨ ⎩
∞
0
yα−1
∞
0
lnxyλ
1xyαfxdx
2
dy
⎫ ⎬ ⎭
1/2
.
3.3
It follows from3.3that the inequality3.1is valid after some simplifications.
On the other hand, assume that the inequality3.1keeps valid; by applying in turn Cauchy’s inequality and3.1, we have
∞
0
lnxyλ 1xyαfxg
ydx dy
∞
0
yα−1/2
∞
0
lnxyλ
1xyαfxdx
yα−1/2gydy
≤ ⎧ ⎨ ⎩ ∞ 0
yα−1
∞
0
lnxyλ
1xyαfxdx
2
dy
⎫ ⎬ ⎭
1/2
∞
0
y1−αg2ydy
1/2
≤
Cα, λ2
∞
0
x1−αf2xdx
1/2∞
0
y1−αg2ydy
1/2
Cα, λ
∞
0
x1−αf2xdx
1/2∞
0
y1−αg2ydy
1/2
.
3.4
Therefore, the inequality3.1is equivalent to2.1.
Theorem 3.2. Letnbe a nonnegative integer andα >0. If0∞x1−αf2xdx <∞, then
∞
0
yα−1
∞
0
lnxy2n
1xyαfxdx
2
dy≤Cα,2n2
∞
0
x1−αf2xdx, 3.5
whereCα,2n π/α2n1En, theEnsare the Euler numbers,namely,E01, E11, E25, E3
61, E41385, and so forth, and the constant factorCα,2nis the best possible. And the equality in
3.5holds if and only iffx 0. And the inequality3.5is equivalent to2.14.
Theorem 3.3. Letfandgbe two real functions, and letαbe a positive numbers. If0∞x1−αf2xdx <
∞and0∞x1−αg2xdx <∞, then
∞
0
yα−1
∞
0
lnxy
1xyαfxdx
2
dy≤ 8G α2
∞
0
x1−αf2xdx
1/2∞
0
x1−αg2xdx
1/2
,
3.6
whereG is the Catalan constant, that is,G 0,915965594· · ·. And the constant factor 8G/α2 in
3.6is the best possible. And the equality in3.6holds if and only iffx 0. And the inequality
3.6is equivalent to2.19.
The proofs of Theorems3.2and3.3are similar to one of Theorem3.1; they are omitted here.
Similarly, we can establish also some new inequalities which they are, respectively, equivalent to the inequalities2.17,2.18and2.20. These are omitted here.
Acknowledgments
Authors would like to express their thanks to the referees for their valuable comments and suggestions. The research is supported by Scientific Research Fund of Hunan Provincial Education Department09C789.
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