Rev. Anal. Num´er. Th´eor. Approx., vol. 34 (2005) no. 1, pp. 71–78 ictp.acad.ro/jnaat
SOME REMARKS ON HILBERT’S INTEGRAL INEQUALITY
LJ. MARANGUNI ´C∗and J. PE ˇCARI ´C†
Abstract. A generalization of the well-known Hilbert’s inequality is given. Sev- eral other results of this type obtained in recent years follow as a special case of our result.
MSC 2000. 26D15.
Keywords. Hilbert’s inequality, H¨older’s inequality, Beta function.
1. INTRODUCTION
First, let us recall the well known Hilbert’s integral inequality:
Theorem A. If f, g∈L2(0,∞), then the following inequality holds:
(1.1)
∞
Z
0
∞
Z
0
f(x)g(y)
x+y dxdy≤π
∞
Z
0
f2(x) dx
∞
Z
0
g2(x) dx
1 2
,
where π is the best constant.
In the recent years a lot of results with generalization of this type of in- equality were obtained. Let us mention two of them which take our attention.
Theorem B. (Gavrea, [1]) i) Let a be a real number such that a > 1. If f, g∈L21a, a, then:
(1.2)
a
Z
1 a
a
Z
1 a
f(x)g(y)
(x+y)λ dxdy≤Ka(λ)
a
Z
1 a
x1−λf2(x) dx
a
Z
1 a
x1−λg2(x) dx
1 2
,
where:
Ka(λ) =
a
Z
1 a
xλ−22
(1+x)λdx, λ >0.
∗Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, Zagreb, Croatia, e-mail: [email protected].
†Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, Zagreb, Croatia, e-mail: [email protected].
ii) Let 0< a < b. If f, g∈L2(a, b), then:
(1.3)
b
Z
a b
Z
a
f(x)g(y)
(x+y)λ dxdy≤Kpb a
(λ)
b
Z
a
x1−λf2(x) dx
b
Z
a
x1−λg2(x) dx
1 2
.
Theorem C (Brneti´c and Peˇcari´c, [2]). If f, g are real functions and λ, p, q >0, 1p+1q = 1, then the following inequalities hold and are equivalent:
(1.4)
∞
Z
0
f(x)g(y)
(x+y)λ dxdy < Bλ2,λ2
∞
Z
0
xp−1−pλ2 fp(x) dx
1 p
∞
Z
0
xq−1−qλ2 gq(x) dx
1 q
and
(1.5)
∞
Z
0
yλp2 −1
∞
Z
0
f(x) dx (x+y)λ
p
dy < Bpλ2,λ2
∞
Z
0
xp−1−pλ2 fp(x) dx, where B is a beta-function.
In this paper we generalize inequalities (1.2) and (1.3). As a special case of our results we obtain Theorem C.
In the rest of our paper we suppose that all integrals converge.
2. MAIN RESULTS
A generalization of inequalities (1.2) and (1.3) is given in the following theorem:
Theorem 1. Let 0 < a < b. If f, g are real functions and p, q > 0,
1
p +1q = 1, then:
(2.1)
b
Z
a b
Z
a
f(x)g(y)
(x+y)λ dxdy≤Kpb a
(λ)
b
Z
a
xp−1−pλ2 fp(x) dx
1 p
b
Z
a
xq−1−qλ2 gq(x) dx
1 q
,
where
Kpb a
(λ) = pb
a
Z
√a b
xλ−22
(1+x)λdx, λ∈R+.
Proof. Our proof consists of two steps. In the first step we prove the fol- lowing lemma:
Lemma1. Let abe a real number such that a >1. Iff, gare real functions and p, q >0, 1p +1q = 1, then:
(2.2)
a
Z
1 a
a
Z
1 a
f(x)g(y)
(x+y)λ dxdy≤Ka(λ)
a
Z
1 a
xp−1−pλ2 fp(x) dx
1 p
a
Z
1 a
xq−1−qλ2 gq(x) dx
1 q
,
where
Ka(λ) =
a
Z
1 a
xλ−22
(1+x)λdx, λ∈R+. Proof. We start with the following equality:
(2.3)
a
Z
1 a
a
Z
1 a
f(x)g(y)
(x+y)λ dxdy =
a
Z
1 a
a
Z
1 a
f(x)x
2−λ 2q y
2−λ 2p
(x+y)λp
·
g(y)y
2−λ 2p x
2−λ 2q
(x+y)λq
dxdy.
By H¨older’s inequality and (2.3), we have:
a
Z
1 a
a
Z
1 a
f(x)g(y)
(x+y)λ dxdy ≤
≤
a
Z
1 a
a
Z
1 a
fp(x) (x+y)λ ·x
p(2−λ) 2q
y2−λ2 dxdy
1 p
a
Z
1 a
a
Z
1 a
gq(y) (x+y)λ ·y
q(2−λ) 2p
x2−λ2 dxdy
1 q
=
a
Z
1 a
a
Z
1 a
fp(x) (x+y)λ ·x
p(2−λ) 2q ·x2−λ2 x2−λ2 ·y2−λ2
dxdy
1 p
×
×
a
Z
1 a
a
Z
1 a
gq(y) (x+y)λ ·y
q(2−λ) 2p ·y2−λ2 y2−λ2 ·x2−λ2
dxdy
1 q
=
a
Z
1 a
fp(x)x
(2−λ)(p−q) 2q
a
Z
1 a
y
x
λ−2 2
(x+y)λdy
dx
1 p
×
×
Za
1 a
gq(y)y
(2−λ)(q−p) 2p
Za
1 a
x y
λ−22 (x+y)λ dx
dy
1 q
=
Za
1 a
fp(x)x
(2−λ)(p−q)
2q Ixdx
1 p
Za
1 a
gq(y)y
(2−λ)(q−p)
2p Iydy
1 q
, (2.4)
where we denoteIx = Z a
1 a
y
x
λ−2
2 (x+y)−λdy,Iy = Z a
1 a
x
y
λ−2
2 (x+y)−λdx.
Using the change of variables y=xt, dy=xdt we have forIx: Ix=
a
Zx
1 ax
tλ−22
(x+xt)λxdt=x1−λ
a
Zx
1 ax
tλ−22
(1 +t)λdt=x1−λh(x) and similarly: Iy =y1−λh(y).
Since x∈1a, a we conclude thath(x) = Z a
x 1 ax
tλ−22 (1 +t)−λdt≥0, and for λ >0 hstrictly increasing on a1,1and strictly decreasing on (1, a) (see also [1]). Hence:
(2.5) h(x)≤h(1) =
a
Z
1 a
tλ−22
(1 +t)λ dt=Ka(λ).
Using this inequality, (2.4) can be rewritten as:
a
Z
1 a
a
Z
1 a
f(x)g(y)
(x+y)λ dxdy≤
≤
a
Z
1 a
fp(x)x
(2−λ)(p−q)
2q +1−λ
h(1) dx
1 p
a
Z
1 a
gq(y)y
(2−λ)(q−p)
2p +1−λ
h(1) dy
1 q
=Ka(λ)
a
Z
1 a
fp(x)xp−1−pλ2 dx
1 p
a
Z
1 a
gq(y)yq−1−qλ2 dy
1 q
and the Lemma is proved.
Remark1. Inequality (2.2) is a generalization of Theorem B i). We obtain
(1.2) by puttingp=q = 2 in (2.2).
We continue proving our Theorem 1 by rearranging (2.2). We put qab instead a,b > a, and obtain:
pb
a
Z
√a
b
pb
a
Z
√a
b
f1(x1)g1(y1)
(x1+y1)λ dx1dy1 ≤ (2.6)
≤Kpb a
(λ)
pb
a
Z
√a b
f1p(x1)xp−1−
pλ 2
1 dx1
1 p
pb
a
Z
√a b
gq1(x1)xq−1−
qλ 2
1 dx1
1 q
.
Now we transform X1OY1 → XOY by putting x = x1√
ab, y = y1√ ab.
Taking into account
∂x1
∂x
∂y1
∂x1 ∂x
∂y
∂y1
∂y
=
√1
ab 0
0 √1
ab
= ab1,qab√
ab=a, qb
a
√ ab=b, we obtain:
b
Z
a b
Z
a
f1√x
ab
g1√y
ab
√x
ab +√y
ab
λ ·dxdy ab ≤
≤Kpb a
(λ)
Zb
a
xp−1−pλ2
(ab)12(p−1−pλ2 )f1p√x
ab
dx
√ab
1 p
×
×
b
Z
a
xq−1−qλ2
(ab)12(q−1−qλ2 )gq1√x
ab
dx
√ ab
1 q
. (2.7)
Replacingf1√x
ab
withf(x) andg1√y
ab
withg(y) we rewrite inequality (2.7):
b
Z
a b
Z
a
f(x)g(y)
(x+y)λ ·(ab)λ2
ab dxdy≤
≤Kpb
a
(λ)
Zb
a
xp−1−pλ2 fp(x)(ab)−12(p−1−pλ2 )
√
ab dx
1 p
×
×
Zb
a
xq−1−qλ2 gq(x)(ab)−12(q−1−qλ2)
√
ab dx
1 q
.
By dividing both sides of the last inequality with
(ab)λ2−1 = (ab)−12+2p1+pλ4p−2p1−12+2q1+qλ4q−2q1
we obtain (2.1), thus completing the proof of Theorem 1.
Remark2. Inequality (2.1) is a generalization of Theorem B ii). We obtain
(1.3) by puttingp=q = 2 in (2.1).
A generalization of inequalities (1.4) and (1.5) is given in the following theorem:
Theorem 2. Let 0< a < b. If f is a real function and p >1, then:
(2.8)
b
Z
a
yλp2 −1
b
Z
a
f(x) dx (x+y)λ
p
dy≤Kpp b
a
(λ)
b
Z
a
xp−1−pλ2 fp(x) dx.
Also, (2.1)and (2.8)are equivalent.
Proof. Let us show that (2.1) and (2.8) are equivalent.
First suppose that inequality (2.1) is valid. Denoting g(y) =
b
Z
a
f(x) dx (x+y)λ
p−1
yλp2 −1 we have:
Zb
a
yλp2−1
Zb
a
f(x) dx (x+y)λ
p
dy=
=
b
Z
a
yλp2−1
b
Z
a
f(x) dx (x+y)λ
p−1 b
Z
a
f(x) dx (x+y)λdy
=
b
Z
a
g(y)
b
Z
a
f(x) dx (x+y)λdy=
b
Z
a b
Z
a
f(x)g(y) (x+y)λ dxdy
≤Kpb a
(λ)
b
Z
a
xp−1−pλ2 fp(x) dx
1 p
b
Z
a
yq−1−qλ2 gq(y) dy
1 q
=Kpb
a
(λ)
Zb
a
xp−1−pλ2 fp(x) dx
1 p
×
×
b
Z
a
yq−1−qλ2 yλpq2 −q
b
Z
a
f(x) dx (x+y)λ
(p−1)q
dy
1 q
=Kpb a
(λ)
b
Z
a
xp−1−pλ2 fp(x) dx
1 p
×
×
b
Z
a
yλp2−1
b
Z
a
f(x) dx (x+y)λ
p+q−q
dy
1 q
.
By putting:
I =
b
Z
a
yλp2−1
b
Z
a
f(x) dx (x+y)λ
p
dy, we can write the last inequality in the following form:
I ≤Kpb a
(λ)
b
Z
a
xp−1−pλ2 fp(x) dx
1 p
×I1q,
wherefrom we have (2.8).
Now let us suppose that inequality (2.8) is valid. By applying H¨older’s inequality (in one variable) and (2.8) we have:
b
Z
a b
Z
a
f(x)g(y)
(x+y)λ dxdy=
=
b
Z
a
y−
q−1−qλ 2 q
b
Z
a
f(x) dx (x+y)λ
y
q−1−qλ 2
q g(y) dy
=
b
Z
a
b
Z
a
y−
q−1−qλ 2
q f(x)
(x+y)λdx
y
q−1−qλ 2
q g(y) dy
≤
b
Z
a
y−
p(q−1−qλ2)
q
b
Z
a
f(x) dx (x+y)λ
p
dy
1 p
b
Z
a
yq−1−qλ2 gq(y) dy
1 q
=
b
Z
a
yλp2 −1
b
Z
a
f(x) dx (x+y)λ
p
dy
1 p
b
Z
a
yq−1−qλ2 gq(y) dy
1 q
,
wherefrom we have (2.1). Since (2.1) has already been proved, the inequality
(2.8) holds, too.
Remark3. Fora→0,b→ ∞we haveKpb a
(λ)→Bλ2,λ2and we obtain
Theorem C as a special case of our result.
REFERENCES
[1] Gavrea, I.,Some Remarks on Hilbert’s Integral Inequality, Mathematical Inequalities &
Applications,3, pp. 473–477, 2002.
[2] Brneti´c, I. and Peˇcari´c, J., Generalizations of inequalities of Hardy-Hilbert type, Mathematical Inequalities & Applications,7, no. 2, pp. 217–225, 2004.
Received by the editors: November 25, 2004.
