A REFINEMENT OF HÖLDER’S INEQUALITY AND
APPLICATIONS
XUEMEI GAO, MINGZHE GAO AND
XIAOZHOU SHANG
Department of Mathematics and Computer Science
Normal College, Jishou University
Jishou Hunan 416000,
People’s Republic of China
EMail: mingzhegao@163.com
Received:
10 February, 2007
Accepted:
10 June, 2007
Communicated by:
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
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S.S. Dragomir
J
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2000 AMS Sub. Class.:
26D15, 46C99.
Page 1 of 19
Key words:
Inner product space, Gram matrix, Variable unit-vector, Minkowski’s inequality,
Fan Ky’s inequality, Hardy’s inequality.
Abstract:
In this paper, it is shown that a refinement of Hölder’s inequality can be established using the positive definiteness of the Gram matrix. As applications, some
improvements on Minkowski’s inequality, Fan Ky’s inequality and Hardy’s inequality are given.
Acknowledgements:
The authors would like to thank the anonymous referee for valuable comments
that have been implemented in the final version of this paper.
A Project Supported by scientific Research Fund of Hunan Provincial Education
Department (06C657).
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Contents
1
Introduction
3
2
Main Results
5
3
Applications
3.1 A Refinement of Minkowski’s Inequality . . . . . . . . . . . . . . .
3.2 A Strengthening of Fan Ky’s Inequality . . . . . . . . . . . . . . .
3.3 An Improvement of Hardy’s Inequality . . . . . . . . . . . . . . . .
9
9
12
15
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
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1.
Introduction
For convenience, we need to introduce the following notations which will be frequently used throughout the paper:
(ar , bs ) =
∞
X
arn bsn ,
kakr =
n=1
r
s
Z
(f , g ) =
arn
Z
s
f (x) g (x) dx,
0
and
! r1
,
kak2 = kak ,
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
n=1
∞
r
∞
X
∞
kf kr =
r
f (x) dx
r1
Xiaozhou Shang
,
kf k2 = kf k ,
vol. 8, iss. 2, art. 44, 2007
0
Sr (α, y) = αr/2 , y kαk−r/2
,
r
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where a = (a1 , a2 , . . . ) are sequences of real numbers, f : [0, ∞) → [0, ∞) are
measurable functions and α and y are elements of an inner product space E of real
sequences.
Let a = (a1 , a2 , . . . ) and b = (b1 , b2 , . . . ) be sequences of real numbers in Rn .
Then Hölder’s inequality can be written in the form:
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(a, b) ≤ kakp kbkq .
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The equality in (1.1) holds if and only if api = kbqi , i = 1, 2, . . . , where k is a
constant.
This inequality is important in function theory, functional analysis, Fourier analysis and analytic number theory, etc. However, there are drawbacks in this inequality.
For example, let
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(1.1)
a = (a1 , a2 , . . . , an , 0, . . . , 0) ,
b = (0, 0, . . . , bn+1 , bn+2 , . . . , b2n ) ,
a, b ∈ R2n .
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If we let ai = bj = 1, i = 1, 2, . . . , n; j = n + 1, n + 2, . . . , 2n, and substitute them
into (1.1), then we have 0 ≤ n. In this case, Hölder’s inequality is meaningless.
In the present paper we establish a new inequality that improves Hölder’s inequality and remedies the defect pointed out above. At the same time, some significant
refinements for a number of the classical inequalities can be established. As space is
limited, only several applications of the new inequality are given.
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
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2.
Main Results
Let α and β be elements of an inner product space E. Then the inner
p product of α
and β is denoted by (α, β) and the norm of α is given by kαk = (α, α). In our
previous papers ([1], [2]), the following result has been obtained by means of the
positive definiteness of the Gram matrix.
Lemma 2.1. Let α, β and γ be three arbitrary vectors of E. If kγk = 1, then
(2.1)
|(α, β)|2 ≤ kαk2 kβk2 − (kαk |x| − kβk |y|)2 ,
where x = (β, γ) , y = (α, γ). The equality in (2.1) holds if and only if α and β are
linearly dependent, or γ is a linear combination of α and β, and xy = 0 but x and y
are not simultaneously equal to zero.
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
Title Page
For the sake of completeness, we give here a short proof of (2.1), which can also
be found in [2].
Proof of Lemma 2.1. Consider the Gram determinant constructed by the vectors α, β
and γ:
(α, α) (α, β) (α, γ) G(α, β, γ) = (β, α) (β, β) (β, γ) .
(γ, α) (γ, β) (γ, γ) According to the positive definiteness of Gram matrix we have G(α, β, γ) ≥ 0, and
G(α, β, γ) = 0 if and only if the vectors α, β and γ are linearly dependent.
Expanding this determinant and using the condition kγk = 1 we obtain
G(α, β, γ) = kαk2 kβk2 − (α, β)2 − kαk2 x2 − 2(α, β)xy + kβk2 y 2
≤ kαk2 kβk2 − (α, β)2 − kαk2 x2 − 2 |(α, β)xy| + kβk2 y 2
≤ kαk2 kβk2 − (α, β)2 − {kαk |x| − kβk |y|}2
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where x = (β, γ) and y = (α, γ). It follows that the equality holds if and only if the
vectors α and β are linearly dependent; or the vector γ is a linear combination of the
vector α and β, and xy = 0 but x and y are not simultaneously equal to zero.
Applying Lemma 2.1, we can now establish the following refinement of Hölder’s
inequality.
Theorem 2.2. Let an, bn ≥ 0, (n = 1, 2, . . . ),
+∞ and 0 < kbkq < +∞, then
1
p
1
q
+
= 1 and p > 1. If 0 < kakp <
Xiaozhou Shang
m
(a, b) ≤ kakp kbkq (1 − r) ,
(2.2)
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
vol. 8, iss. 2, art. 44, 2007
where
2
r = (Sp (a, c) − Sq (b, c)) ,
m = min
1 1
,
p q
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,
kck = 1
and
ap/2 , c
bq/2 , c ≥ 0.
The equality in (2.2) holds if and only if ap/2 and bq/2 are linearly
or if
dependent;
the vector c is a linear combination of ap/2 and bq/2 , and ap/2 , c bq/2 , c = 0, but
the vector c is not simultaneously orthogonal to ap/2 and bq/2 .
Proof. Firstly, we consider the case p 6= q. Without loss of generality, we suppose
p
that p > q > 1. Since p1 + 1q = 1, we have p > 2. Let R = p2 , Q = p−2
, then
1
1
+ Q = 1. By Hölder’s inequality we obtain
R
(2.3)
(a, b) =
∞
X
k=1
ak b k =
∞ X
k=1
q/p
ak b k
1−q/p
bk
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≤
∞ X
q/p
ak b k
R
k=1
= a
p/2
! R1
∞ X
1−q/p
bk
Q
! Q1
k=1
q/2 2/p
,b
kbkq(1−2/p)
q
.
The equality in (2.3) holds if and only if ap/2 and bq/2 are linearly dependent. In fact,
the equality in (2.3) holds if and only if for any k, there exists c0 (c0 6= 0) such that
R
Q
1−q/p
q/p
ak b k
= c 0 bk
.
p/2
q/2
It is easy to deduce that ak = c0 bk .
If α, β and γ in (2.1) are replaced by ap/2 , bq/2 and c respectively, then we have
2
(2.4)
ap/2 , bq/2 ≤ kakpp kbkqq (1 − r) ,
where r = (Sp (a, c) − Sq (b, c))2 . Substituting (2.4) into (2.3), we obtain after simplifications
(2.5)
1
(a, b) ≤ kakp kbkq (1 − r) p .
It is known from Lemma 2.1 that the equality in (2.5) holds if and only if ap/2 and
bq/2 are linearly
dependent;
or if the vector c is a linear combination of ap/2 and bq/2 ,
and ap/2 , c bq/2 , c = 0, but the vector c is not simultaneously orthogonal to ap/2
and bq/2 .
Note the symmetry of p and q. The inequality (2.2) follows from (2.5).
Secondly, consider the case p = 2. By Lemma 2.1, we obtain:
(2.6)
1
(a, b) ≤ kak kbk (1 − r) 2 ,
2
(b,c)
where r = (a,c)
−
, kck = 1 and (a, c) (b, c) ≥ 0. The equality in (2.6) holds
kak
kbk
if and only if a and b are linearly dependent, or the vector c is a linear combination
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
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of a and b, and (a, c)(b, c) = 0, but (a, c) and (b, c) are not simultaneously equal to
zero.
The proof of the theorem is thus completed.
Consider the example given in the Introduction. Let c = (c1 , c2 , . . . , c2n ), c ∈
R , where ci = √1n , i = 1, 2, . . . , n and cj = 0, j = n + 1, n + 2, . . . , 2n. It is easy
to deduce that kck = 1 and r = 1. Substituting them into (2.2), it follows that the
equality is valid.
The following theorem provides a similar result to Theorem 2.2.
2n
Theorem 2.3. Let f (x) , g (x) ≥ 0 (x ∈ (0, +∞)),
0 < kf kp < +∞ and 0 < kgkq < +∞, then
1
p
+
1
q
= 1 and p > 1. If
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
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(f, g) ≤ kf kp kgkq (1 − r)m ,
(2.7)
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Contents
where
2
r = (Sp (f, h) − Sq (g, h)) ,
m = min
Z
khk = 1,
i.e.
khk =
∞
h2 (x)dx
1 1
,
p q
,
12
=1
0
and
f p/2 , h g q/2 , h ≥ 0.
The equality in (2.3) holds if and only if f p/2 and g q/2 are linearly
dependent;
or the
q/2
p/2
q/2
p/2
vector h is a linear combination of f and g , and f , h g , h = 0, but the
vector h is not simultaneously orthogonal to f p/2 and g q/2 .
Its proof is similar to that of Theorem 2.2. Hence it is omitted.
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3.
Applications
3.1.
A Refinement of Minkowski’s Inequality
We firstly give a refinement of Minkowski’s inequality for the discrete form.
Theorem 3.1. Let ak , bk ≥ 0, p > 1. If 0 < kakp < +∞ and 0 < kbkp < +∞, then
(3.1)
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
ka + bkp < kakp + kbkp (1 − r)m ,
Xiaozhou Shang
where
vol. 8, iss. 2, art. 44, 2007
ka + bkp =
∞
X
! p1
(ak + bk )p
,
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k=1
1
1
,1 −
p
p
r = min {r (a) , r (b)} ,
m = min
,
(
)2
xp/2 , c
(a + b)p/2 , c
r (x) =
−
, x = a, b;
kxkp/2
ka + bkp/2
p
p
∞
X
p/2
(a + b) , c =
(ak + bk )p/2 ck ,
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k=1
and c is a variable unit-vector.
n
o
Proof. Let m = min p1 , 1 − p1 ,
ka + bkp =
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∞
X
k=1
! p1
p
(ak + bk )
.
By Theorem 2.2, we have
(3.2)
∞
X
!1− p1
∞
X
ak (ak + bk )p−1 ≤ kakp
k=1
k=1
∞
X
∞
X
(1 − r(a))m
(ak + bk )p
and
(3.3)
bk (ak + bk )p−1 ≤ kbkp
!1− p1
(ak + bk )p
(1 − r(b))m ,
k=1
k=1
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
where
(
r (x) =
xp/2 , c
kxkp/2
p
−
(a + b)p/2 , c
(a + b)
,c =
∞
X
Title Page
,
ka + bkp/2
p
ka + bkp/2
=
p
p/2
)2
x = a, b,
! 12
(ak + bk )p
k=1
∞
X
,
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p/2
(ak + bk )
ck ,
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k=1
and c is a variable unit-vector.
Adding (3.5) and (3.3) we obtain, after simplifying:
(3.4)
ka + bkp ≤ kakp (1 − r(a))m + kbkp (1 − r(b))m .
Let r = min {r (a) , r (b)}, then the inequality (3.1) follows. This completes the
proof of Theorem 3.1.
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If we choose a unit-vector c such that its ith component is 1 and the rest is zero,
i.e. c = (0, 0, . . . , 0, 1 , 0, . . . ), then
(i)
(
r (x) =
p/2
xi
kxkp/2
p
−
(ai + bi )p/2
)2
x = a, b.
ka + bkp/2
p
Similarly, we can establish a refinement of Minkowski’s integral inequality.
Theorem 3.2. Let f (x) , g (x) ≥ 0, p > 1. If 0 < kf kp < +∞ and 0 < kgkp <
+∞, then
(3.5)
kf + gkp < kf kp + kgkp (1 − r)m ,
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
Title Page
where
∞
Z
p
kf + gkp =
(f (x) + g (x)) dx ,
1
1
r = min {r (f ) , r (g)} , m = min
,1 −
,
p
p
(
)2
(f + g)p/2 , h
tp/2 , h
r (t) =
−
, t = f, g,
p/2
ktkp/2
kf
+
gk
p
p
Z ∞
(f + g)p/2 , h =
(f (x) + g (x))p/2 h (x) dx,
0
0
and h is a variable unit-vector, i.e.
Z
khk =
0
∞
2
h (x) dx
12
= 1.
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Its proof is similar to that of Theorem 3.1. Hence it is omitted.
Remark 1. The variable unit-vector h can be chosen in accordance with our requirements. For example, we may choose h such that
s
2
h (x) =
.
π (1 + x2 )
3.2.
A Strengthening of Fan Ky’s Inequality
Xiaozhou Shang
Theorem 3.3. Let A, B and C be three positive definite matrices of order n, 0 ≤
λ ≤ 1. Then
(3.6)
vol. 8, iss. 2, art. 44, 2007
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|A|λ |B|1−λ


1
4
1
4
m = min {λ, 1 − λ}.
Proof. When λ = 0, 1, the inequality (3.3) is obviously valid. Hence we need only
consider the case 0 < λ < 1.
If D is a positive definite matrix of order n, then it is known from [4] that
Z +∞
Z +∞
π n/2
(3.7)
Jn =
···
e−(x,Dx) dx =
1 ,
−∞
−∞
|D| 2
where x = (x1 , x2 , . . . , xn ), and dx = dx1 dx2 · · · dxn .
Contents
2 m
|BC|
|AC|
≤ |λA + (1 − λ) B| 1 −  1 − 1   ,
1 (B + C) 2
1 (A + C) 2
2
2
where |C| = π n ,
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Let F (x) = e−λ(x,Ax) and G (x) = e−(1−λ)(x,Bx) . If p =
ing to (3.4) and (2.7) we have
1
λ
and q =
1
,
1−λ
accord-
π n/2
(3.8)
1
|λA + (1 − λ) B| 2
Z +∞
Z +∞
=
···
F (x) G (x) dx
−∞
Z
+∞
≤
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
−∞
Z
···
−∞
n/2
p1 Z
+∞
p
+∞
Z
+∞
q
···
F (x) dx
−∞
m
−∞
1q
G (x) dx
(1 − r)m
−∞
π (1 − r)
=
1 ,
λ
1−λ 2
|A| |B|
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
Title Page
Contents
where
2
r = S 1 (F, H) − S 1 (G, H)
λ
1−λ
1
1
1
− 2(1−λ)
1
− 2λ
2(1−λ)
=
F 2λ , H kF k 1 − G
, H kGk 1
,
λ
1−λ
1
where H = e− 2 (x,Cx) , C is a positive definite matrix of order n, and
Z
+∞
kHk =
Z
+∞
···
−∞
2
H (x) dx
21
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= 1.
−∞
By the definition of the variable unit-vector H, it is easy to deduce that |C| = π n .
Hence we have
1
2λ
F
Z
+∞
Z
+∞
···
,H =
−∞
1
F 2λ (x) H (x) dx
−∞
(
π n/2
=
1 =
1 (A + C) 2
2
|C|
1
(A + C)
2
) 12
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
and
Xiaozhou Shang
1/2λ
kF k1/λ
Z
+∞
Z
···
=
−∞
21
+∞
F
1/λ
(x) dx
(
=
π n/2
) 12
|A|1/2
−∞
=
|C|
|A|
14
vol. 8, iss. 2, art. 44, 2007
,
Title Page
whence
Contents
1
|AC| 4
S1/λ (F, H) = 1 .
1 (A + C) 2
2
Similarly,
1
4
|BC|
S1/(1−λ) (G, H) = 1 ,
1 (B + C) 2
2
therefore we obtain
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
(3.9)
JJ
|AC|
1
4
|BC|
1
4
2
r = 
1 − 1
1  .
1 (A + C) 2
(B + C) 2
2
2
It follows from (3.8) and (3.9) that the inequality (3.3) is valid.
Close
3.3.
An Improvement of Hardy’s Inequality
We give firstly a refinement of Hardy’s inequality for the discrete form.
P
Theorem 3.4. Let an ≥ 0, βn = n1 nk=1 ak , p1 + 1q = 1 and p > 1. If 0 < kakp <
+∞, then
p
kakp (1 − r)m ,
(3.10)
kβkp ≤
p−1
Xiaozhou Shang
where
r=
(ap/2 , c)
kakp/2
p
−
(β p/2 , c)
kβkp/2
p
n
o
c is a variable unit-vector and m = min p1 , 1q .
!2
vol. 8, iss. 2, art. 44, 2007
,
Proof. Firstly, we estimate the difference of the following two terms:
(3.11)
βnp −
p
p
βnp−1 an = βnp −
(nβn − (n − 1)βn−1 ) βnp−1
p−1
p−1
p1
(n − 1)p
np
p
p
= βn 1 −
+
(βnp )p−1 βn−1
.
p−1
p−1
Applying the arithmetic-geometric mean inequality to the second term on the righthand side of (3.11) we get
(3.12)
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Xuemei Gao, Mingzhe Gao and
p
(βnp )p−1 βn−1
p1
≤
1
p
(p − 1)βnp + βn−1
.
p
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It follows from (3.11) and (3.12) that
p
np
(n − 1)
p
p
p−1
p
βn −
βn an ≤ βn 1 −
+
(p − 1)βnp + βn−1
p−1
p−1
p−1
1
p
=
(n − 1)βn−1
− nβnp .
p−1
Summing the above inequality with respect to n, we have
N
X
n=1
N
βnp
1
p X p−1
p
βn an ≤ −
−
(N βN
) ≤ 0.
p − 1 n=1
p−1
Hence
N
X
N
βnp
n=1
p X p−1
≤
β an .
p − 1 n=1 n
Letting N → ∞, we get
(3.13)
∞
X
n=1
∞
βnp
p X p−1
≤
β an .
p − 1 n=1 n
Applying the inequality (2.2) to the right-hand side of (3.13) we obtain
! p1 ∞
! 1q
∞
∞
X
X
p X
p
an βnp−1 ≤
ap
(3.14)
βn(p−1)q
(1 − r)m
p − 1 n=1
p − 1 n=1 n
n=1
1q
p
=
kakp kβkpp (1 − r)m ,
p−1
n
o
2
1 1
p−1
where r = (Sp (a, c) − Sq (β , c)) , c is a variable unit-vector and m = min p , q .
Hölder’s Inequality
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We obtain from (3.13) and (3.14) after simplification
p
kakp (1 − r)m .
(3.15)
kβkp ≤
p−1
It is easy to deduce that
ap/2 , c
Sp (a, c) =
kakp/2
p
Sq (β p−1 , c) =
and
(β (p−1)q/2 , c)
kβ p−1 kq/2
q
=
(β p/2 , c)
kβkp/2
p
.
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
Hence
r=
a
p/2
,c
kakp−p/2
− β
p/2
,c
kβk−p/2
p
2
vol. 8, iss. 2, art. 44, 2007
,
where c is a variable unit-vector. The proof of the theorem is completed.
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A variable unit-vector c can be chosen in accordance with our requirements. For
example, we may choose c ∈ R∞ such that c = (1, 0, 0, . . . ). Obviously, kck = 1
and
2
−p/2
r = ap1 kak−p/2
−
kβk
.
p
p
Contents
Similarly, we can establish a refinement of Hardy’s integral inequality.
Rx
Theorem 3.5. Let f (x) ≥ 0, g(x) = x1 0 f (t)dt, p1 + 1q = 1 and p > 1. If
R∞
0 < 0 f (t)dt < +∞, then
(3.16)
kgkp <
p
kf kp (1 − r)m ,
p−1
where
r=
(f p/2 , h)
kf kp/2
p
−
(g p/2 , h)
kgkp/2
p
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!2
,
h is a variable unit-vector, i.e.
Z ∞
12
1 1
2
khk =
h (t)dt
= 1 and m = min
,
.
p q
0
Proof. Using integration by parts and then applying (2.2) we obtain that
Z ∞
p
p
(3.17)
f, g p−1
kgkp =
g p (t)dt =
p−1
0
p−1 p
≤
kf kp g q (1 − r)m
p−1
p
=
kf kp kgkp−1
(1 − r)m ,
p
p−1
o
n
2
where r = (Sp (f, h) − Sq (g p−1 , h)) , m = min p1 , 1q and h is a variable unitvector. It is easy to deduce that
g p/2 , h
f p/2 , h
p−1
Sp (f, h) =
and
Sq g , h =
.
p/2
kf kp/2
kgk
p
p
It follows that the inequality (3.16) is valid. The theorem is thus proved.
A variable unit-vector h can be chosen in accordance with our requirements. For
example, we may choose h such that h (x) = e−x/2 . Obviously, we then have
Z
khk =
0
∞
12
h2 (t)dt
= 1.
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
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References
[1] MINGZHE GAO, On Heisenberg’s inequality, J. Math. Anal. Appl., 234(2)
(1999), 727–734.
[2] TIAN-XIAO HE, J.S. SHIUE AND ZHONGKAI LI, Analysis, Combinatorics
and Computing, Nova Science Publishers, Inc. New York, 2002, 197-204.
[3] JICHANG KUANG, Applied Inequalities, Hunan Education Press, 2nd ed.
1993. MR 95j: 26001.
[4] E.F. BECKENBACK
1965.
AND
R. BELLMAN, Inequalities, 2nd ed., Springer,
Hölder’s Inequality
and Applications
Xuemei Gao, Mingzhe Gao and
Xiaozhou Shang
vol. 8, iss. 2, art. 44, 2007
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