Journal of Inequalities in Pure and
Applied Mathematics
REFINEMENTS OF THE SCHWARZ AND HEISENBERG
INEQUALITIES IN HILBERT SPACES
volume 5, issue 3, article 60,
2004.
S.S. DRAGOMIR
School of Computer Science and Mathematics
Victoria University of Technology
PO Box 14428
Melbourne VIC 8001
Australia.
EMail: sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Received 24 August, 2004;
accepted 17 September, 2004.
Communicated by: S. Saitoh
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
156-04
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Abstract
Some new refinements of the Schwarz inequality in inner product spaces are
given. Applications for discrete and integral inequalities including the Heisenberg inequality for vector-valued functions in Hilbert spaces are provided.
2000 Mathematics Subject Classification: Primary 46C05; Secondary 26D15.
Key words: Schwarz inequality, Triangle inequality, Heisenberg Inequality.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Some New Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Discrete Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4
Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5
Refinements of Heisenberg Inequality . . . . . . . . . . . . . . . . . . . 26
References
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
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1.
Introduction
Let (H; h·, ·i) be an inner product space over the real or complex number field
K. One of the most important inequalities in inner product spaces with numerous applications, is the Schwarz inequality
(1.1)
|hx, yi|2 ≤ kxk2 kyk2 ,
x, y ∈ H
with equality iff x and y are linearly dependent.
In 1966, S. Kurepa [1] established the following refinement of the Schwarz
inequality in inner product spaces that generalises de Bruijn’s result for sequences of real and complex numbers [2].
Theorem 1.1. Let H be a real Hilbert space and HC the complexification of H.
Then for any pair of vectors a ∈ H, z ∈ HC
(1.2)
|hz, ai|2 ≤
1
kak2 kzk2 + |hz, z̄i| ≤ kak2 kzk2 .
2
In 1985, S.S. Dragomir [3, Theorem 2] obtained a different refinement of
(1.1), namely:
Theorem 1.2. Let (H; h·, ·i) be a real or complex inner product space and
x, y, e ∈ H with kek = 1. Then we have the inequality
(1.3)
kxk kyk ≥ |hx, yi − hx, ei he, yi| + |hx, ei he, yi| ≥ |hx, yi| .
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
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In the same paper [3, Theorem 3], a further generalisation for orthonormal
families has been given (see also [4, Theorem 3]).
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Theorem 1.3. Let {ei }i∈H be an orthonormal family in the Hilbert space H.
Then for any x, y ∈ H
X
X
kxk kyk ≥ hx, yi −
(1.4)
hx, ei i hei , yi +
|hx, ei i hei , yi|
i∈I
i∈I
X
X
≥ hx, yi −
hx, ei i hei , yi + hx, ei i hei , yi
i∈I
i∈I
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
≥ |hx, yi| .
The inequality (1.3) has also been obtained in [4] as a particular case of the
following result.
Theorem 1.4. Let x, y, a, b ∈ H be such that
kak2 ≤ 2 Re hx, ai ,
kbk2 ≤ 2 Re hy, bi .
Then we have:
(1.5)
1
1
kxk kyk ≥ 2 Re hx, ai − kak2 2 2 Re hy, bi − kbk2 2
+ |hx, yi − hx, bi − ha, yi + ha, bi| .
Another refinement of the Schwarz inequality for orthornormal vectors in
inner product spaces has been obtained by S.S. Dragomir and J. Sándor in [5,
Theorem 5].
S.S. Dragomir
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Theorem 1.5. Let {ei }i∈{1,...,n} be orthornormal vectors in the inner product
space (H; h·, ·i). Then
(1.6)
kxk kyk − |hx, yi|
≥
n
X
n
X
|hx, ei i|2
|hy, ei i|2
i=1
i=1
! 12
n
X
−
hx, ei i hei , yi ≥ 0
i=1
and
(1.7)
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
kxk kyk − Re hx, yi
≥
n
X
n
X
2
|hx, ei i|
|hy, ei i|2
i=1
i=1
! 12
−
n
X
S.S. Dragomir
Re [hx, ei i hei , yi] ≥ 0.
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i=1
For some properties of superadditivity, monotonicity, strong superadditivity
and strong monotonicity of Schwarz’s inequality, see [6]. Here we note only
the following refinements of the Schwarz inequality in its different variants for
linear operators [6]:
a) Let H be a Hilbert space and A, B : H → H two selfadjoint linear
operators with A ≥ B ≥ 0, then we have the inequalities
(1.8)
1
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1
hAx, xi 2 hAy, yi 2 − |hAx, yi|
1
1
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≥ hBx, xi 2 hBy, yi 2 − |hBx, yi| ≥ 0
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and
(1.9) hAx, xi hAy, yi − |hAx, yi|2 ≥ hBx, xi hBy, yi − |hBx, yi|2 ≥ 0
for any x, y ∈ H.
b) Let A : H → H be a bounded linear operator on H and let kAk =
sup {kAxk , kxk = 1} the norm of A. Then one has the inequalities
(1.10)
kAk2 (kxk kyk − |hx, yi|) ≥ kAxk kAyk − |hAx, Ayi| ≥ 0
and
(1.11) kAk4
kxk2 kyk2 − |hx, yi|2 ≥ kAxk2 kAyk2 −|hAx, Ayi|2 ≥ 0.
c) Let B : H → H be a linear operator with the property that there exists a
constant m > 0 such that kBxk ≥ m kxk for any x ∈ H. Then we have
the inequalities
(1.12)
2
kBxk kByk − |hBx, Byi| ≥ m (kxk kyk − |hx, yi|) ≥ 0
and
2
2
2
2
2
2
(1.13) kBxk kByk − |hBx, Byi| ≥ m4 kxk kyk − |hx, yi|
≥ 0.
For other results related to Schwarz’s inequality in inner product spaces, see
Chapter XX of [8] and the references therein.
Motivated by the results outlined above, it is the aim of this paper to explore
other avenues in obtaining new refinements of the celebrated Schwarz inequality. Applications for vector-valued sequences and integrals in Hilbert spaces are
mentioned. Refinements of the Heisenberg inequality for vector-valued functions in Hilbert spaces are also given.
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
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2.
Some New Refinements
The following result holds.
Theorem 2.1. Let (H; h·, ·i) be an inner product space over the real or complex
number field K and r1 , r2 > 0. If x, y ∈ H satisfy the property
kx − yk ≥ r2 ≥ r1 ≥ |kxk − kyk| ,
(2.1)
then we have the following refinement of Schwarz’s inequality
kxk kyk − Re hx, yi ≥
(2.2)
1 2
r2 − r12 (≥ 0) .
2
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
1
2
The constant is best possible in the sense that it cannot be replaced by any
larger quantity.
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Proof. From the first inequality in (2.1) we have
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(2.3)
kxk2 + kyk2 ≥ r22 + 2 Re hx, yi .
Subtracting in (2.3) the quantity 2 kxk kyk , we get
(2.4)
(kxk − kyk)2 ≥ r22 − 2 (kxk kyk − Re hx, yi) .
Since, by the second inequality in (2.1) we have
(2.5)
r12 ≥ (kxk − kyk)2 ,
hence from (2.4) and (2.5) we deduce the desired inequality (2.2).
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To prove the sharpness of the constant
a constant C > 0 such that
(2.6)
1
2
in (2.2), let us assume that there is
kxk kyk − Re hx, yi ≥ C r22 − r12 ,
provided that x and y satisfy (2.1).
Let e ∈ H with kek = 1 and for r2 > r1 > 0, define
(2.7)
x=
r 2 + r1
r1 − r2
· e and y =
· e.
2
2
Then
kx − yk = r2 and |kxk − kyk| = r1 ,
showing that the condition (2.1) is fulfilled with equality.
If we replace x and y as defined in (2.7) into the inequality (2.6), then we get
r22 − r12
≥ C r22 − r12 ,
2
which implies that C ≤ 12 , and the theorem is completely proved.
The following corollary holds.
Corollary 2.2. With the assumptions of Theorem 2.1, we have the inequality:
√
√ q
2
2
(2.8)
kxk + kyk −
kx + yk ≥
r22 − r12 .
2
2
Refinements of the Schwarz
and Heisenberg Inequalities in
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S.S. Dragomir
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Proof. We have, by (2.2), that
(kxk + kyk)2 − kx + yk2 = 2 (kxk kyk − Re hx, yi) ≥ r22 − r12 ≥ 0
which gives
(2.9)
2
2
(kxk + kyk) ≥ kx + yk +
2
q
r22
−
r12
.
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
By making use of the elementary inequality
2 α2 + β 2 ≥ (α + β)2 ,
α, β ≥ 0;
S.S. Dragomir
we get
(2.10)
2
kx + yk +
2
q
r22
−
r12
1
≥
2
kx + yk +
q
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2
r22
−
r12
.
Utilising (2.9) and (2.10), we deduce the desired inequality (2.8).
If (H; h·, ·i) is a Hilbert space and {ei }i∈I is an orthornormal family in H,
i.e., we recall that hei , ej i = δij for any i, j ∈ I, where δij is Kronecker’s delta,
then we have the following inequality which is well known in the literature as
Bessel’s inequality
X
(2.11)
|hx, ei i|2 ≤ kxk2 for each x ∈ H.
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i∈I
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Here, the meaning of the sum is
(
)
X
X
2
2
|hx, ei i| = sup
|hx, ei i| , F is a finite part of I .
F ⊂I
i∈I
i∈F
The following result providing a refinement of the Bessel inequality (2.11)
holds.
Theorem 2.3. Let (H; h·, ·i) be a Hilbert space and {ei }i∈I an orthornormal
family in H. If x ∈ H, x 6= 0, and r2 , r1 > 0 are such that:
! 12
X
X
(2.12) x −
(≥ 0) ,
hx, ei i ei ≥ r2 ≥ r1 ≥ kxk −
|hx, ei i|2
i∈I
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! 12
kxk −
X
|hx, ei i|2
≥
i∈I
The constant
1
·
2 P
r22
i∈I
−
r12
1
2 2
(≥ 0) .
|hx, ei i|
1
2
is best possible.
P
Proof. Consider y :=
i∈I hx, ei i ei . Obviously, since H is a Hilbert space,
y ∈ H. We also note that
v
2 s
X
u
X
X
u
t
kyk = hx, ei i ei = hx, ei i ei =
|hx, ei i|2 ,
i∈I
S.S. Dragomir
i∈I
then we have the inequality
(2.13)
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
i∈I
i∈I
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and thus (2.12) is in fact (2.1) of Theorem 2.1.
Since
! 21
*
+
X
X
2
kxk kyk − Re hx, yi = kxk
|hx, ei i|
− Re x,
hx, ei i ei
i∈I
i∈I
! 21 
=
X
|hx, ei i|2
kxk −
i∈I
! 12 
X
|hx, ei i|2
,
i∈I
hence, by (2.2), we deduce the desired result (2.13).
We will prove the sharpness of the constant for the case of one element, i.e.,
I = {1} , e1 = e ∈ H, kek = 1. For this, assume that there exists a constant
D > 0 such that
r2 − r12
kxk − |hx, ei| ≥ D · 2
|hx, ei|
(2.14)
provided x ∈ H\ {0} satisfies the condition
(2.15)
kx − hx, ei ek ≥ r2 ≥ r1 ≥ kxk − |hx, ei| .
Refinements of the Schwarz
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S.S. Dragomir
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Assume that x = λe + µf with e, f ∈ H, kek = kf k = 1 and e ⊥ f. We wish
to see if there exists positive numbers λ, µ such that
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(2.16)
kx − hx, ei ek = r2 > r1 = kxk − |hx, ei| .
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Since (for λ, µ > 0)
kx − hx, ei ek = µ
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and
kxk − |hx, ei| =
p
λ2 + µ2 − λ
hence, by (2.16), we get µ = r2 and
q
λ2 + r22 − λ = r1
giving
λ2 + r22 = λ2 + 2λr1 + r12
from where we get
r22 − r12
> 0.
2r1
With these values for λ and µ, we have
λ=
kxk − |hx, ei| = r1 ,
r2 − r12
|hx, ei| = 2
2r1
giving D ≤
1
.
2
This proves the theorem.
The following corollary is obvious.
r22 − r12
r22 −r12
2r1
S.S. Dragomir
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and thus, from (2.14), we deduce
r1 ≥ D ·
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Corollary 2.4. Let x, y ∈ H with hx, yi =
6 0 and r2 ≥ r1 > 0 such that
hx, yi (2.17)
kyk x − kyk · y ≥ r2 kyk ≥ r1 kyk
≥ kxk kyk − |hx, yi| (≥ 0) .
Then we have the following refinement of the Schwarz’s inequality:
kyk2
1 2
2
kxk kyk − |hx, yi| ≥
r − r1
(≥ 0) .
2 2
|hx, yi|
(2.18)
The constant
1
2
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
is best possible.
S.S. Dragomir
The following lemma holds.
Lemma 2.5. Let (H; h·, ·i) be an inner product space and R ≥ 1. For x, y ∈ H,
the subsequent statements are equivalent:
(i) The following refinement of the triangle inequality holds:
(2.19)
kxk + kyk ≥ R kx + yk ;
(ii) The following refinement of the Schwarz inequality holds:
(2.20)
kxk kyk − Re hx, yi ≥
1 2
R − 1 kx + yk2 .
2
Proof. Taking the square in (2.19), we have
(2.21)
2 kxk kyk ≥ R2 − 1 kxk2 + 2R2 Re hx, yi + R2 − 1 kyk2 .
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Subtracting from both sides of (2.21) the quantity 2 Re hx, yi , we obtain
2 (kxk kyk − Re hx, yi) ≥ R2 − 1 kxk2 + 2 Re hx, yi + kyk2
= R2 − 1 kx + yk2 ,
which is clearly equivalent to (2.20).
By the use of the above lemma, we may now state the following theorem
concerning another refinement of the Schwarz inequality.
Theorem 2.6. Let (H; h·, ·i) be an inner product space over the real or complex
number field and R ≥ 1, r ≥ 0. If x, y ∈ H are such that
(2.22)
1
(kxk + kyk) ≥ kx + yk ≥ r,
R
then we have the following refinement of the Schwarz inequality
(2.23)
kxk kyk − Re hx, yi ≥
1 2
R − 1 r2 .
2
The constant 21 is best possible in the sense that it cannot be replaced by a larger
quantity.
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
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Proof. The inequality (2.23) follows easily from Lemma 2.5. We need only
prove that 21 is the best possible constant in (2.23).
Assume that there exists a C > 0 such that
(2.24)
kxk kyk − Re hx, yi ≥ C R2 − 1 r2
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provided x, y, R and r satisfy (2.22).
Consider r = 1, R > 1 and choose x = 1−R
e, y = 1+R
e with e ∈ H,
2
2
kek = 1. Then
kxk + kyk
x + y = e,
=1
R
and thus (2.22) holds with equality on both sides.
From (2.24), for the above choices, we have 12 (R2 − 1) ≥ C (R2 − 1) ,
which shows that C ≤ 21 .
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
Finally, the following result also holds.
Theorem 2.7. Let (H; h·, ·i) be an inner product space over the real or complex
number field K and r ∈ (0, 1]. For x, y ∈ H, the following statements are
equivalent:
S.S. Dragomir
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(i) We have the inequality
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|kxk − kyk| ≤ r kx − yk ;
(2.25)
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(ii) We have the following refinement of the Schwarz inequality
(2.26)
The constant
kxk kyk − Re hx, yi ≥
1
2
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1
1 − r2 kx − yk2 .
2
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in (2.26) is best possible.
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Proof. Taking the square in (2.25), we have
kxk2 − 2 kxk kyk + kyk2 ≤ r2 kxk2 − 2 Re hx, yi + kyk2
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which is clearly equivalent to
1 − r2 kxk2 − 2 Re hx, yi + kyk2 ≤ 2 (kxk kyk − Re hx, yi)
or with (2.26).
Now, assume that (2.26) holds with a constant E > 0, i.e.,
(2.27)
kxk kyk − Re hx, yi ≥ E 1 − r2 kx − yk2 ,
provided (2.25) holds.
Define x = r+1
e, y =
2
r−1
e
2
with e ∈ H, kek = 1. Then
|kxk − kyk| = r,
kx − yk = 1
showing that (2.25) holds with equality.
If we replace x and y in (2.27), then we get E (1 − r2 ) ≤ 21 (1 − r2 ) , implying that E ≤ 12 .
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S.S. Dragomir
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3.
Discrete Inequalities
Assume that (K; (·, ·)) is a Hilbert space
P∞over the real or complex number field.
Assume also that pi ≥ 0, i ∈ H with i=1 pi = 1 and define
(
)
∞
X
`2p (K) := x := (xi ) xi ∈ K, i ∈ N and
pi kxi k2 < ∞ .
i∈N
i=1
It is well known that `2p (K) endowed with the inner product h·, ·ip defined by
hx, yip :=
∞
X
pi (xi , yi )
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
i=1
and generating the norm
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kxkp :=
∞
X
! 12
pi kxi k
2
i=1
is a Hilbert space over K.
We may state the following discrete inequality improving the CauchyBunyakovsky-Schwarz classical result.
Proposition
3.1. Let (K; (·, ·)) be a Hilbert space and pi ≥ 0 (i ∈ N) with
P∞
2
i=1 pi = 1. Assume that x, y ∈ `p (K) and r1 , r2 > 0 satisfy the condition
(3.1)
kxi − yi k ≥ r2 ≥ r1 ≥ |kxi k − kyi k|
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for each i ∈ N. Then we have the following refinement of the Cauchy-BunyakovskySchwarz inequality
(3.2)
∞
X
∞
X
pi kxi k2
pi kyi k2
i=1
i=1
The constant
1
2
! 21
−
∞
X
pi Re (xi , yi ) ≥
i=1
1 2
r2 − r12 ≥ 0.
2
is best possible.
Proof. From the condition (3.1) we simply deduce
(3.3)
∞
X
2
pi kxi − yi k ≥
r22
≥
r12
i=1
≥
∞
X
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
pi (kxi k − kyi k)2
S.S. Dragomir
i=1

≥
∞
X
! 12
pi kxi k
2
i=1
−
∞
X
! 1 2
2
2
pi kyi k
 .
i=1
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In terms of the norm k·kp , the inequality (3.3) may be written as
(3.4)
kx − ykp ≥ r2 ≥ r1 ≥ kxkp − kykp .
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Utilising Theorem 2.1 for the Hilbert space `2p (K) , h·, ·ip , we deduce the
desired inequality (3.2).
For n = 1 (p1 = 1) , the inequality (3.2) reduces to (2.2) for which we have
shown that 21 is the best possible constant.
By the use of Corollary 2.2, we may state the following result as well.
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Corollary 3.2. With the assumptions of Proposition 3.1, we have the inequality
(3.5)
∞
X
! 21
2
pi kxi k
+
i=1
∞
X
! 12
pi kyi k
2
i=1
√
2
−
2
∞
X
! 12
pi kxi + yi k
i=1
2
√ q
2
≥
r22 − r12 .
2
The following proposition also holds.
Proposition
3.3. Let (K; (·, ·)) be a Hilbert space and pi ≥ 0 (i ∈ N) with
P∞
p
=
1.
Assume that x, y ∈ `2p (K) and R ≥ 1, r ≥ 0 satisfy the condition
i=1 i
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
1
(3.6)
(kxi k + kyi k) ≥ kxi + yi k ≥ r
R
for each i ∈ N. Then we have the following refinement of the Schwarz inequality
(3.7)
∞
X
pi kxi k
i=1
∞
X
2
! 21
pi kyi k
2
−
∞
X
i=1
i=1
1 2
pi Re (xi , yi ) ≥
R − 1 r2 .
2
1
2
The constant is best possible in the sense that it cannot be replaced by a larger
quantity.
Proof. By (3.6) we deduce
"∞
# 21
X
1
pi (kxi k + kyi k)2 ≥
(3.8)
R i=1
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∞
X
i=1
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! 12
2
pi kxi + yi k
≥ r.
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By the classical Minkowsky inequality for nonnegative numbers, we have
(3.9)
∞
X
i=1
! 12
pi kxi k2
+
∞
X
i=1
! 21
pi kyi k2
"
≥
∞
X
# 12
pi (kxi k + kyi k)2
,
i=1
and thus, by utilising (3.8) and (3.9), we may state in terms of k·kp the following
inequality
(3.10)
1
kxkp + kykp ≥ kx + ykp ≥ r.
R
Employing Theorem 2.6 for the Hilbert space `2p (K) and the inequality (3.10),
we deduce the desired result (3.7).
Since, for p = 1, n = 1, (3.7) is reduced to (2.23) for which we have shown
that 21 is the best constant, we conclude that 12 is the best constant in (3.7) as
well.
Finally, we may state and prove the following result incorporated in
Proposition
3.4. Let (K; (·, ·)) be a Hilbert space and pi ≥ 0 (i ∈ N) with
P∞
p
=
1.
Assume that x, y ∈ `2p (K) and r ∈ (0, 1] such that
i=1 i
(3.11)
|kxi k − kyi k| ≤ r kxi − yi k for each i ∈ N,
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
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holds true. Then we have the following refinement of the Schwarz inequality
(3.12)
∞
X
pi kxi k
i=1
2
∞
X
! 12
pi kyi k
2
−
i=1
∞
X
pi Re (xi , yi )
i=1
∞
≥
The constant
1
2
X
1
1 − r2
pi kxi − yi k2 .
2
i=1
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
is best possible in (3.12).
Proof. From (3.11) we have
S.S. Dragomir
"
∞
X
i=1
# 12
pi (kxi k − kyi k)2
"
≤r
∞
X
# 12
pi kxi − yi k2
.
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i=1
Utilising the following elementary result
! 21
! 12 ∞
∞
X
X
2
2
≤
p
kx
k
−
p
ky
k
i
i
i
i
i=1
i=1
Contents
∞
X
! 12
pi (kxi k − kyi k)2
,
i=1
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we may state that
kxkp − kykp ≤ r kx − ykp .
Now, by making use of Theorem 2.7, we deduce the desired inequality (3.12)
and the fact that 21 is the best possible constant. We omit the details.
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4.
Integral Inequalities
Assume that (K; (·, ·)) is a Hilbert space over the real or complex number
field K. If ρ : [a, b] ⊂ R → [0, ∞) is a Lebesgue integrable function with
Rb
ρ (t) dt = 1, then we may consider the space L2ρ ([a, b] ; K) of all functions
a
Rb
f : [a, b] → K, that are Bochner measurable and a ρ (t) kf (t)k2 dt < ∞. It is
known that L2ρ ([a, b] ; K) endowed with the inner product h·, ·iρ defined by
b
Z
hf, giρ :=
ρ (t) (f (t) , g (t)) dt
a
and generating the norm
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
Z
kf kρ :=
b
21
ρ (t) kf (t)k dt
2
a
is a Hilbert space over K.
Now we may state and prove the first refinement of the Cauchy-BunyakovskySchwarz integral inequality.
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Proposition 4.1. Assume that f, g ∈ L2ρ ([a, b] ; K) and r2 , r1 > 0 satisfy the
condition
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kf (t) − g (t)k ≥ r2 ≥ r1 ≥ |kf (t)k − kg (t)k|
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for a.e. t ∈ [a, b] . Then we have the inequality
b
Z
2
Z
b
21
ρ (t) kg (t)k dt
2
ρ (t) kf (t)k dt
a
Z b
1 2
−
ρ (t) Re (f (t) , g (t)) dt ≥
r2 − r12 (≥ 0) .
2
a
(4.2)
a
The constant
1
2
is best possible in (4.2).
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
Proof. Integrating (4.1), we get
b
Z
(4.3)
12
ρ (t) (kf (t) − g (t)k)2 dt
S.S. Dragomir
a
Z
≥ r2 ≥ r 1 ≥
b
12
ρ (t) (kf (t)k − kg (t)k) dt .
2
a
Utilising the obvious fact
Z
(4.4)
a
b
1
2
ρ (t) (kf (t)k − kg (t)k) dt
Z
12 12 Z b
b
≥
ρ (t) kf (t)k2 dt
−
ρ (t) kg (t)k2 dt ,
a
a
2
we can state the following inequality in terms of the k·kρ norm:
(4.5)
kf − gkρ ≥ r2 ≥ r1 ≥ kf kρ − kgkρ .
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Employing Theorem 2.1 for the Hilbert space L2ρ ([a, b] ; K) , we deduce the
desired inequality (4.2).
To prove the sharpness of 12 in (4.2), we choose a = 0, b = 1, f (t) = 1,
t ∈ [0, 1] and f (t) = x, g (t) = y, t ∈ [a, b] , x, y ∈ K. Then (4.2) becomes
kxk kyk − Re hx, yi ≥
1 2
r2 − r12
2
provided
kx − yk ≥ r2 ≥ r1 ≥ |kxk − kyk| ,
which, by Theorem 2.1 has the quantity
1
2
as the best possible constant.
The following corollary holds.
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
Corollary 4.2. With the assumptions of Proposition 4.1, we have the inequality
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Z
(4.6)
a
b
12 Z b
12
2
2
ρ (t) kf (t)k dt
+
ρ (t) kg (t)k dt
a
√ Z b
√ q
12
2
2
2
−
ρ (t) kf (t) + g (t)k dt
≥
r22 − r12 .
2
2
a
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The following two refinements of the Cauchy-Bunyakovsky-Schwarz (CBS)
integral inequality also hold.
Proposition 4.3. If f, g ∈ L2ρ ([a, b] ; K) and R ≥ 1, r ≥ 0 satisfy the condition
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(4.7)
1
(kf (t)k + kg (t)k) ≥ kf (t) + g (t)k ≥ r
R
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for a.e. t ∈ [a, b] , then we have the inequality
Z
b
(4.8)
a
21
ρ (t) kf (t)k2 dt
ρ (t) kg (t)k2 dt
a
Z b
1 2
−
ρ (t) Re (f (t) , g (t)) dt ≥
R − 1 r2 .
2
a
Z
1
2
The constant
b
is best possible in (4.8).
The proof follows by Theorem 2.6 and we omit the details.
Proposition 4.4. If f, g ∈ L2ρ ([a, b] ; K) and ζ ∈ (0, 1] satisfy the condition
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
|kf (t)k − kg (t)k| ≤ ζ kf (t) − g (t)k
(4.9)
for a.e. t ∈ [a, b] , then we have the inequality
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Z
(4.10)
a
The constant
b
Z
12
b
ρ (t) kg (t)k2 dt
ρ (t) kf (t)k2 dt
a
Z b
−
ρ (t) Re (f (t) , g (t)) dt
a
Z
b
1
2
1−ζ
ρ (t) kf (t) − g (t)k2 dt.
≥
2
a
1
2
is best possible in (4.10).
The proof follows by Theorem 2.7 and we omit the details.
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5.
Refinements of Heisenberg Inequality
It is well known that if (H; h·, ·i) is a real or complex Hilbert space and f :
[a, b] ⊂ R →H is an absolutely continuous vector-valued function, then f is
differentiable almost everywhere on [a, b] , the derivative f 0 : [a, b] → H is
Bochner integrable on [a, b] and
Z t
(5.1)
f (t) =
f 0 (s) ds
for any t ∈ [a, b] .
a
The following theorem provides a version of the Heisenberg inequalities in
the general setting of Hilbert spaces.
Theorem 5.1. Let ϕ : [a, b] → H be an absolutely continuous function with the
property that b kϕ (b)k2 = a kϕ (a)k2 . Then we have the inequality:
Z
(5.2)
a
2
Z b
Z b
b
2
2
2
2
kϕ (t)k dt ≤ 4
t kϕ (t)k dt ·
kϕ0 (t)k dt.
a
a
The constant 4 is best possible in the sense that it cannot be replaced by any
smaller constant.
Proof. Integrating by parts, we have successively
b Z b
Z b
d
2
2
(5.3)
kϕ (t)k dt = t kϕ (t)k −
t
kϕ (t)k2 dt
dt
a
a
a
Z b
d
2
2
= b kϕ (b)k − a kϕ (a)k −
t hϕ (t) , ϕ (t)i dt
dt
a
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
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Z
b
=−
t [hϕ0 (t) , ϕ (t)i + hϕ (t) , ϕ0 (t)i] dt
a
Z
b
= −2
t Re hϕ0 (t) , ϕ (t)i dt
a
Z b
=2
Re hϕ0 (t) , (−t) ϕ (t)i dt.
a
If we apply the Cauchy-Bunyakovsky-Schwarz integral inequality
Z
b
Z
Re hg (t) , h (t)i dt ≤
b
kg (t)k2 dt
a
a
Z
21
b
2
kh (t)k dt
a
for g (t) = ϕ0 (t) , h (t) = −tϕ (t) , t ∈ [a, b] , then we deduce the desired
inequality (4.5).
The fact that 4 is the best constant in (4.5) follows from the fact that in the
(CBS) inequality, the case of equality holds iff g (t) = λh (t) for a.e. t ∈ [a, b]
and λ a given scalar in K. We omit the details.
For details on the classical Heisenberg inequality, see, for instance, [7].
Utilising Proposition 4.1, we can state the following refinement of the Heisenberg inequality obtained above in (5.2):
Proposition 5.2. Assume that ϕ : [a, b] → H is as in the hypothesis of Theorem
5.1. In addition, if there exist r2 , r1 > 0 so that
0
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S.S. Dragomir
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0
kϕ (t) + tϕ (t)k ≥ r2 ≥ r1 ≥ |kϕ (t)k − |t| kϕ (t)k|
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for a.e. t ∈ [a, b] , then we have the inequality
Z
b
2
2
Z
b
t kϕ (t)k dt ·
a
21
Z
1 b
kϕ (t)k dt
−
kϕ (t)k2 dt
2 a
1
≥ (b − a) r22 − r12 (≥ 0) .
2
0
a
2
The proof follows by Proposition 4.1 on choosing f (t) = ϕ0 (t) , g (t) =
1
−tϕ (t) and ρ (t) = b−a
, t ∈ [a, b] .
On utilising Proposition 4.3 for the same choices of f, g and ρ, we may state
the following results as well:
Proposition 5.3. Assume that ϕ : [a, b] → H is as in the hypothesis of Theorem
5.1. In addition, if there exist R ≥ 1 and r > 0 so that
1
(kϕ0 (t)k + |t| kϕ (t)k) ≥ kϕ0 (t) − tϕ (t)k ≥ r
R
for a.e. t ∈ [a, b] , then we have the inequality
Z
b
t2 kϕ (t)k2 dt ·
Z
a
a
b
12
Z
1 b
0
kϕ (t)k dt
−
kϕ (t)k2 dt
2 a
1
≥ (b − a) R2 − 1 r2 (≥ 0) .
2
2
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
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Finally, we can state
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Proposition 5.4. Let ϕ : [a, b] → H be as in the hypothesis of Theorem 5.1. In
addition, if there exists ζ ∈ (0, 1] so that
|kϕ0 (t)k − |t| kϕ (t)k| ≤ ζ kϕ0 (t) + tϕ (t)k
for a.e. t ∈ [a, b] , then we have the inequality
Z
b
2
2
Z
t kϕ (t)k dt ·
a
a
b
21
Z
1 b
kϕ (t)k2 dt
kϕ (t)k dt
−
2 a
Z
b 0
1
2
2
kϕ (t) + tϕ (t)k dt (≥ 0) .
1−ζ
≥
2
a
0
2
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
This follows by Proposition 4.4 and we omit the details.
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References
[1] S. KUREPA, On the Buniakowsky-Cauchy-Schwarz inequality, Glasnik
Mat. Ser III, 1(21) (1966), 147–158.
[2] N.G. DE BRUIJN, Problem 12, Wisk. Opgaven, 21 (1960), 12–13.
[3] S.S. DRAGOMIR, Some refinements of Schwarz inequality, Suppozionul
de Matematică şi Aplicaţii, Polytechnical Institute Timişoara, Romania, 1-2
November 1985, 13–16.
[4] S.S. DRAGOMIR AND J. SÁNDOR, Some inequalities in prehilbertian
spaces, Studia Univ., Babeş-Bolyai, Mathematica, 32(1) (1987), 71–78 MR
89h: 46034.
[5] S.S. DRAGOMIR AND J. SÁNDOR, On Bessels’ and Gram’s inequalities
in prehilbertian spaces, Periodica Math. Hungarica, 29(3) (1994), 197–
205.
[6] S.S. DRAGOMIR AND B. MOND, On the superadditivity and monotonicity
of Schwarz’s inequality in inner product spaces, Contributions, Macedonian
Acad. of Sci. and Arts, 15(2) (1994), 5–22.
Refinements of the Schwarz
and Heisenberg Inequalities in
Hilbert Spaces
S.S. Dragomir
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[7] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge University Press, Cambridge, United Kingdom, 1952.
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[8] D.S. MITRINOVĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and
New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.
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