Journal of Inequalities in Pure and
Applied Mathematics
REVERSES OF SCHWARZ, TRIANGLE AND BESSEL
INEQUALITIES IN INNER PRODUCT SPACES
volume 5, issue 3, article 76,
2004.
S.S. DRAGOMIR
School of Computer Science and Mathematics
Victoria University of Technology
PO Box 14428, MCMC 8001
Victoria, Australia.
EMail: sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Received 12 December, 2003;
accepted 11 July, 2004.
Communicated by: B. Mond
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
171-03
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Abstract
Reverses of the Schwarz, triangle and Bessel inequalities in inner product
spaces that improve some earlier results are pointed out. They are applied
to obtain new Grüss type inequalities. Some natural integral inequalities are
also stated.
2000 Mathematics Subject Classification: Primary 26D15, 46C05.
Key words: Schwarz’s inequality, Triangle inequality, Bessel’s inequality, Grüss type
inequalities, Integral inequalities.
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Some Reverses of Schwarz’s Inequality . . . . . . . . . . . . . . . . . .
3
Reverses of the Triangle Inequality . . . . . . . . . . . . . . . . . . . . . .
4
Some Grüss Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Reverses of Bessel’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . .
6
Some Grüss Type Inequalities for Orthonormal Families . . . .
7
Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
S.S. Dragomir
3
9
16
19
23
29
33
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1.
Introduction
Let (H; h·, ·i) be an inner product space over the real or complex number field
K. The following inequality is known in the literature as Schwarz’s inequality:
(1.1)
|hx, yi|2 ≤ kxk2 kyk2 ,
x, y ∈ H;
where kzk2 = hz, zi , z ∈ H. The equality occurs in (1.1) if and only if x and y
are linearly dependent.
In [7], the following reverse of Schwarz’s inequality has been obtained:
(1.2)
0 ≤ kxk2 kyk2 − |hx, yi|2 ≤
1
|A − a|2 kyk4 ,
4
provided x, y ∈ H and a, A ∈ K are so that either
(1.3)
Re hAy − x, x − ayi ≥ 0,
or, equivalently,
(1.4)
x − a + A · y ≤ 1 |A − a| kyk ,
2
2
holds. The constant 14 is best possible in (1.2) in the sense that it cannot be
replaced by a smaller quantity.
If x, y, A, a satisfy either (1.3) or (1.4), then the following reverse of Schwarz’s
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
S.S. Dragomir
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inequality also holds [8]
h
i
Re
Ahx,
yi
+
a
hx,
yi
1
kxk kyk ≤ ·
1
2
[Re (aA)] 2
(1.5)
≤
1 |A| + |a|
·
|hx, yi| ,
2 [Re (aA)] 12
provided that, the complex numbers a and A satisfy the condition Re (aA) > 0.
In both inequalities in (1.5), the constant 12 is best possible.
An additive version of (1.5) may be stated as well (see also [9])
(1.6)
0 ≤ kxk2 kyk2 − |hx, yi|2
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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S.S. Dragomir
2
≤
1 (|A| − |a|) + 4 [|Aa| − Re (aA)]
|hx, yi|2 .
·
Re (aA)
4
In this inequality, 41 is the best possible constant.
It has been proven in [10], that
(1.7)
2
φ + ϕ
1
2
2
2
0 ≤ kxk − |hx, yi| ≤ |φ − ϕ| − − hx, ei ;
4
2
provided, either
(1.8)
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Re hφe − x, x − ϕei ≥ 0,
or, equivalently,
(1.9)
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1
φ
+
ϕ
x −
≤ |φ − ϕ| ,
e
2
2
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where e ∈ H, kek = 1. The constant 14 in (1.7) is also best possible.
y
If we choose e = kyk
, φ = Γ kyk , ϕ = γ kyk (y 6= 0) , Γ, γ ∈ K, then by
(1.8) and (1.9) we have,
Re hΓy − x, x − γyi ≥ 0,
(1.10)
or, equivalently,
(1.11)
x − Γ + γ y ≤ 1 |Γ − γ| kyk ,
2
2
imply the following reverse of Schwarz’s inequality:
(1.12)
2
2
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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S.S. Dragomir
2
0 ≤ kxk kyk − |hx, yi|
2
Γ + γ
1
2
4
2
≤ |Γ − γ| kyk − kyk − hx, yi .
4
2
The constant 14 in (1.12) is sharp.
Note that this inequality is an improvement of (1.2), but it might not be very
convenient for applications.
Now, let {ei }i∈I be a finite or infinite family of orthornormal vectors in the
inner product space (H; h·, ·i) , i.e., we recall that

 0 if i 6= j
hei , ej i =
, i, j ∈ I.

1 if i = j
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In [11], we proved that, if {ei }i∈I is as above, F ⊂ I is a finite part of I such
that either
+
*
X
X
(1.13)
Re
φi ei − x, x −
ϕi ei ≥ 0,
i∈F
i∈F
or, equivalently,
(1.14)
1
X
φ
+
ϕ
i
i e
x
−
i ≤
2
2
i∈F
! 12
X
|φi − ϕi |2
,
i∈F
holds, where (φi )i∈I , (ϕi )i∈I are real or complex numbers, then we have the
following reverse of Bessel’s inequality:
X
0 ≤ kxk2 −
(1.15)
|hx, ei i|2
Reverses of Schwarz, Triangle
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S.S. Dragomir
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i∈F
*
+
X
X
1 X
2
|φi − ϕi | − Re
φi ei − x, x −
ϕi ei
≤ ·
4 i∈F
i∈F
i∈F
1 X
≤ ·
|φi − ϕi |2 .
4 i∈F
The constant 14 in both inequalities is sharp. This result improves an earlier
result by N. Ujević obtained only for real spaces [21].
In [10], by the use of a different technique, another reverse of Bessel’s in-
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equality has been proven, namely:
X
(1.16)
0 ≤ kxk2 −
|hx, ei i|2
i∈F
2
X φi + ϕi
1 X
2
≤ ·
|φi − ϕi | −
−
hx,
e
i
i
2
4 i∈F
i∈F
1 X
≤ ·
|φi − ϕi |2 ,
4 i∈F
provided that (ei )i∈I , (φi )i∈I , (ϕi )i∈I , x and F are as above.
Here the constant 41 is sharp in both inequalities.
It has also been shown that P
the bounds provided by (1.15) and (1.16) for
the Bessel’s difference kxk2 − i∈F |hx, ei i|2 cannot be compared in general,
meaning that there are examples for which one is smaller than the other [10].
Finally, we recall another type of reverse for Bessel inequality that has been
obtained in [12]:
P
2 X
1
2
i∈F (|φi | + |ϕi |)
(1.17)
kxk ≤ · P
|hx, ei i|2 ;
4
Re
(φ
ϕ
)
i i
i∈F
i∈F
provided (φi )i∈I , (ϕi )i∈I satisfy (1.13) (or, equivalently (1.14)) and
0. Here the constant
1
4
is also best possible.
P
i∈F
Re (φi ϕi ) >
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An additive version of (1.17) is
X
(1.18) 0 ≤ kxk2 −
|hx, ei i|2
i∈F
1
≤ ·
4
P
i∈F
(|φi | − |ϕi |)2 + 4 [|φi ϕi | − Re (φi ϕi )] X
P
|hx, ei i|2 .
i∈F Re (φi ϕi )
i∈F
The constant 41 is best possible.
It is the main aim of the present paper to point out new reverse inequalities
to Schwarz’s, triangle and Bessel’s inequalities.
Some results related to Grüss’ inequality in inner product spaces are also
pointed out. Natural applications for integrals are also provided.
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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S.S. Dragomir
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2.
Some Reverses of Schwarz’s Inequality
The following result holds.
Theorem 2.1. Let (H; h·, ·i) be an inner product space over the real or complex
number field K (K = R, K = C) and x, a ∈ H, r > 0 are such that
x ∈ B (a, r) := {z ∈ H| kz − ak ≤ r} .
(2.1)
(i) If kak > r, then we have the inequalities
(2.2) 0 ≤ kxk2 kak2 − |hx, ai|2 ≤ kxk2 kak2 − [Re hx, ai]2 ≤ r2 kxk2 .
The constant 1 in front of r2 is best possible in the sense that it cannot be
replaced by a smaller one.
(ii) If kak = r, then
kxk ≤ 2 Re hx, ai ≤ 2 |hx, ai| .
The constant 2 is best possible in both inequalities.
(iii) If kak < r, then
(2.4)
kxk2 ≤ r2 − kak2 + 2 Re hx, ai ≤ r2 − kak2 + 2 |hx, ai| .
Here the constant 2 is also best possible.
Proof. Since x ∈ B (a, r) , then obviously kx − ak2 ≤ r2 , which is equivalent
to
(2.5)
S.S. Dragomir
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2
(2.3)
Reverses of Schwarz, Triangle
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kxk2 + kak2 − r2 ≤ 2 Re hx, ai .
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(i) If kak > r, then we may divide (2.5) by
(2.6)
q
kak2 − r2 > 0 getting
kxk2
q
2 Re hx, ai
q
+ kak2 − r2 ≤ q
.
kak2 − r2
kak2 − r2
Using the elementary inequality
αp +
1
√
q ≥ 2 pq, α > 0, p, q ≥ 0,
α
we may state that
(2.7)
S.S. Dragomir
kxk2
2 kxk ≤ q
2
kak −
+
q
2
kak −
r2 .
r2
Making use of (2.6) and (2.7), we deduce
q
(2.8)
kxk kak2 − r2 ≤ Re hx, ai .
Taking the square in (2.8) and re-arranging the terms, we deduce the third
inequality in (2.2). The others are obvious.
To prove the sharpness of the constant, assume, under the hypothesis of
the theorem, that, there exists a constant c > 0 such that
(2.9)
kxk2 kak2 − [Re hx, ai]2 ≤ cr2 kxk2 ,
provided x ∈ B (a, r) and kak > r.
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√
Let r = √ε > 0, ε ∈ (0, 1) , a, e ∈ H with kak = kek = 1 and a ⊥ e. Put
x = a + εe. Then obviously x ∈ B (a, r) , kak > r and kxk2 = kak2 +
ε kek2 = 1 + ε, Re hx, ai = kak2 = 1, and thus kxk2 kak2 − [Re hx, ai]2 =
ε. Using (2.9), we may write that
ε ≤ cε (1 + ε) , ε > 0
giving
(2.10)
c + cε ≥ 1 for any ε > 0.
Letting ε → 0+, we get from (2.10) that c ≥ 1, and the sharpness of the
constant is proved.
(ii) The inequality (2.3) is obvious by (2.5) since kak = r. The best constant
follows in a similar way to the above.
Reverses of Schwarz, Triangle
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S.S. Dragomir
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(iii) The inequality (2.3) is obvious. The best constant may be proved in a
similar way to the above. We omit the details.
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The following reverse of Schwarz’s inequality holds.
Theorem 2.2. Let (H; h·, ·i) be an inner product space over K and x, y ∈ H,
γ, Γ ∈ K such that either
(2.11)
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Re hΓy − x, x − γyi ≥ 0,
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or, equivalently,
x − Γ + γ y ≤ 1 |Γ − γ| kyk ,
2
2
(2.12)
holds.
(i) If Re (Γγ) > 0, then we have the inequalities
2
Re Γ + γ hx, yi
1
2
2
(2.13)
kxk kyk ≤ ·
4
Re (Γγ)
2
1 |Γ + γ|
≤ ·
|hx, yi|2 .
4 Re (Γγ)
The constant
1
4
is best possible in both inequalities.
(ii) If Re (Γγ) = 0, then
(2.14)
kxk2 ≤ Re
Reverses of Schwarz, Triangle
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Γ + γ hx, yi ≤ |Γ + γ| |hx, yi| .
(iii) If Re (Γγ) < 0, then
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(2.15)
kxk2 ≤ − Re (Γγ) kyk2 + Re
Γ + γ hx, yi
≤ − Re (Γγ) kyk2 + |Γ + γ| |hx, yi| .
Proof. The proof of the equivalence between the inequalities (2.11) and (2.12)
follows by the fact that in an inner
product
Re hZ − x, x − zi ≥ 0 for
space
≤ 1 kZ − zk (see for example [9]).
x, z, Z ∈ H is equivalent with x − z+Z
2
2
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Consider, for y 6= 0, a =
γ+Γ
y
2
and r = 12 |Γ − γ| kyk . Then
|Γ + γ|2 − |Γ − γ|2
kak − r =
kyk2 = Re (Γγ) kyk2 .
4
2
2
(i) If Re (Γγ) > 0, then the hypothesis of (i) in Theorem 2.1 is satisfied, and
by the second inequality in (2.2) we have
kxk2
2 1
1 |Γ + γ|2
≤ |Γ − γ|2 kxk2 kyk2
kyk2 −
Re Γ + γ hx, yi
4
4
4
from where we derive
2
|Γ + γ|2 − |Γ − γ|2
1 ,
kxk2 kyk2 ≤
Re Γ + γ hx, yi
4
4
giving the first inequality in (2.13).
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The second inequality is obvious.
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1
,
4
To prove the sharpness of the constant assume that the first inequality
in (2.13) holds with a constant c > 0, i.e.,
2
Re
Γ
+
γ
hx,
yi
(2.16)
kxk2 kyk2 ≤ c ·
,
Re (Γγ)
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provided Re (Γγ) > 0 and either (2.11) or (2.12) holds.
Assume that Γ, γ > 0, and let x = γy. Then (2.11) holds and by (2.16) we
deduce
(Γ + γ)2 γ 2 kyk4
4
2
γ kyk ≤ c ·
Γγ
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giving
Γγ ≤ c (Γ + γ)2 for any Γ, γ > 0.
(2.17)
Let ε ∈ (0, 1) and choose in (2.17), Γ = 1 + ε, γ = 1 − ε > 0 to get
1 − ε2 ≤ 4c for any ε ∈ (0, 1) . Letting ε → 0+, we deduce c ≥ 41 , and the
sharpness of the constant is proved.
(ii) and (iii) are obvious and we omit the details.
Remark 2.1. We observe that the second bound in (2.13) for kxk2 kyk2 is better
than the second bound provided by (1.5).
The following corollary provides a reverse inequality for the additive version
of Schwarz’s inequality.
Reverses of Schwarz, Triangle
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Corollary 2.3. With the assumptions of Theorem 2.2 and if Re (Γγ) > 0, then
we have the inequality:
(2.18)
1 |Γ − γ|2
|hx, yi|2 .
0 ≤ kxk kyk − |hx, yi| ≤ ·
4 Re (Γγ)
The constant
1
4
2
2
2
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is best possible in (2.18).
The proof is obvious from (2.13) on subtracting in both sides the same quantity |hx, yi|2 . The sharpness of the constant may be proven in a similar manner
to the one incorporated in the proof of (i), Theorem 2.2. We omit the details.
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Remark 2.2. It is obvious that the inequality (2.18) is better than (1.6) obtained
in [9].
For some recent results in connection to Schwarz’s inequality, see [2], [13]
and [15].
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3.
Reverses of the Triangle Inequality
The following reverse of the triangle inequality holds.
Proposition 3.1. Let (H; h·, ·i) be an inner product space over the real or complex number field K (K = R, C) and x, a ∈ H, r > 0 are such that
kx − ak ≤ r < kak .
(3.1)
Then we have the inequality
(3.2)
0 ≤ kxk + kak − kx + ak
v
u
√
Re hx, ai
u
q
.
≤ 2r · u q
t
2
2
kak − r2
kak − r2 + kak
Proof. Using the inequality (2.8), we may write that
kak Re hx, ai
kxk kak ≤ q
,
2
2
kak − r
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giving
(3.3)
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0 ≤ kxk kak − Re hx, ai
q
kak − kak2 − r2
q
≤
Re hx, ai
2
2
kak − r
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r2 Re hx, ai
q
.
2
2
2
2
kak − r
kak − r + kak
=q
Since
(kxk + kak)2 − kx + ak2 = 2 (kxk kak − Re hx, ai) ,
hence, by (3.3), we have
v
u
u
kxk + kak ≤ ukx + ak2 + q
t
2r2 Re hx, ai
q
2
2
2
2
kak − r
kak − r + kak
≤ kx + ak +
√
v
u
u
2r · u q
t
Re hx, ai
q
,
2
2
2
2
kak − r
kak − r + kak
giving the desired inequality (3.2).
The following proposition providing a simpler reverse for the triangle inequality also holds.
Proposition 3.2. Let (H; h·, ·i) be an inner product space over K and x, y ∈ H,
M ≥ m > 0 such that either
(3.4)
Re hM y − x, x − myi ≥ 0,
or, equivalently,
(3.5)
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1
M
+
m
≤ (M − m) kyk ,
x −
·
y
2
2
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holds. Then we have the inequality
√
(3.6)
0 ≤ kxk + kyk − kx + yk ≤
Proof. Choosing in (2.8), a =
kxk kyk
giving
M +m
y,
2
√
√
M − mp
√
Re hx, yi.
4
mM
r = 12 (M − m) kyk we get
Mm ≤
M +m
Re hx, yi
2
√
√ 2
M− m
√
Re hx, yi .
0 ≤ kxk kyk − Re hx, yi ≤
2 mM
Following the same arguments as in the proof of Proposition 3.1, we deduce the
desired inequality (3.6).
For some results related to triangle inequality in inner product spaces, see
[3], [17], [18] and [19].
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4.
Some Grüss Type Inequalities
We may state the following result.
Theorem 4.1. Let (H; h·, ·i) be an inner product space over the real or complex
number field K (K = R, K = C) and x, y, e ∈ H with kek = 1. If r1 , r2 ∈ (0, 1)
and
kx − ek ≤ r1 ,
(4.1)
ky − ek ≤ r2 ,
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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then we have the inequality
|hx, yi − hx, ei he, yi| ≤ r1 r2 kxk kyk .
S.S. Dragomir
The inequality (4.2) is sharp in the sense that the constant 1 in front of r1 r2
cannot be replaced by a smaller quantity.
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(4.2)
Proof. Apply Schwarz’s inequality in (H; h·, ·i) for the vectors x − hx, ei e,
y − hy, ei e, to get (see also [9])
(4.3)
|hx, yi − hx, ei he, yi|2 ≤ kxk2 − |hx, ei|2 kyk2 − |hy, ei|2 .
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Using Theorem 2.1 for a = e, we may state that
(4.4)
2
2
kxk − |hx, ei| ≤
r12
2
kxk ,
2
Close
2
kyk − |hy, ei| ≤
Utilizing (4.3) and (4.4), we deduce
(4.5)
|hx, yi − hx, ei he, yi|2 ≤ r12 r22 kxk2 kyk2 ,
r22
2
kyk .
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which is clearly equivalent to the desired inequality (4.2).
The sharpness of the constant follows by the fact that for x = y, r1 = r2 = r,
we get from (4.2) that
kxk2 − |hx, ei|2 ≤ r2 kxk2 ,
(4.6)
provided kek = 1 and kx − ek ≤ r < 1. The inequality (4.6) is sharp, as shown
in Theorem 2.1, and the proof is completed.
Another companion of the Grüss inequality may be stated as well.
Theorem 4.2. Let (H; h·, ·i) be an inner product space over K and x, y, e ∈ H
with kek = 1. Suppose also that a, A, b, B ∈ K (K = R, C) such that Re (Aa) ,
Re Bb > 0. If either
(4.7)
Re hAe − x, x − aei ≥ 0, Re hBe − y, y − bei ≥ 0,
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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S.S. Dragomir
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Contents
or, equivalently,
1
a
+
A
x −
≤ |A − a| ,
(4.8)
e
2
2
1
b
+
B
y −
≤ |B − b| ,
e
2
2
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holds, then we have the inequality
(4.9)
1
|A − a| |B − b|
|hx, yi − hx, ei he, yi| ≤ · q
|hx, ei he, yi| .
4
Re (Aa) Re Bb
The constant
1
4
is best possible.
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Proof. We know, by (4.3), that
(4.10)
|hx, yi − hx, ei he, yi|2 ≤ kxk2 − |hx, ei|2
kyk2 − |hy, ei|2 .
If we use Corollary 2.3, then we may state that
kxk2 − |hx, ei|2 ≤
(4.11)
1 |A − a|2
·
|hx, ei|2
4 Re (Aa)
and
1 |B − b|2
|hy, ei|2 .
kyk − |hy, ei| ≤ ·
4 Re Bb
2
(4.12)
2
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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S.S. Dragomir
Utilizing (4.10) – (4.12), we deduce
|hx, yi − hx, ei he, yi|2 ≤
1 |A − a|2 |B − b|2
|hx, ei he, yi|2 ,
·
16 Re (Aa) Re Bb
which is clearly equivalent to the desired inequality (4.9).
The sharpness of the constant follows from Corollary 2.3, and we omit the
details.
Remark 4.1. With the assumptions of Theorem 4.2 and if hx, ei , hy, ei =
6 0
(that is actually the interesting case), then one has the inequality
1
hx, yi
|A − a| |B − b|
(4.13)
.
hx, ei he, yi − 1 ≤ 4 · q
Re (Aa) Re Bb
The constant
1
4
is best possible.
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Remark 4.2. The inequality (4.9) provides a better bound for the quantity
|hx, yi − hx, ei he, yi|
than (2.3) of [9].
For some recent results on Grüss type inequalities in inner product spaces,
see [4], [6] and [20].
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
S.S. Dragomir
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5.
Reverses of Bessel’s Inequality
Let (H; h·, ·i) be a real or complex infinite dimensional Hilbert space and (ei )i∈N
an orthornormal family in H, i.e., we recall that hei , ej i = 0 if i, j ∈ N, i 6= j
and kei k = 1 for i ∈ N.
P
2
It is well known that, if x ∈ H, then the series ∞
i=1 |hx, ei i| is convergent
and the following inequality, called Bessel’s inequality
∞
X
(5.1)
|hx, ei i|2 ≤ kxk2 ,
i=1
holds.
P
2
If `2 (K) := a = (ai )i∈N ⊂ K ∞
i=1 |ai | < ∞ , where K = C or K = R,
is the Hilbert space of all complex or realPsequences that are 2-summable and
λ = (λi )i∈N ∈ `2 (K) , then the series ∞
i=1 λi ei is convergent in H and if
1
P∞
P∞
2 2
.
y := i=1 λi ei ∈ H, then kyk =
i=1 |λi |
We may state the following result.
Theorem 5.1. Let (H; h·, ·i) be an infinite dimensional Hilbert space over the
real or complex number field K, (ei )i∈N an orthornormal family in H, λ =
(λi )i∈N ∈ `2 (K) and r > 0 with the property that
(5.2)
∞
X
2
2
|λi | > r .
i=1
S.S. Dragomir
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If x ∈ H is such that
(5.3)
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∞
X
λi ei ≤ r,
x −
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then we have the inequality
(5.4)
2
i=1 Re λi hx, ei i
P∞
2
2
i=1 |λi | − r
2
P∞
i=1 λi hx, ei i
P∞
2
2
i=1 |λi | − r
P∞
∞
2
X
i=1 |λi |
|hx, ei i|2
P∞
2
2
i=1 |λi | − r i=1
P∞
2
kxk ≤
≤
≤
;
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
and
(5.5)
S.S. Dragomir
0 ≤ kxk2 −
∞
X
|hx, ei i|2
Title Page
i=1
r2
2
2
i=1 |λi | − r
≤ P∞
(5.6)
∞
X
i=1
Proof. Applying the third inequality in (2.2) for a =
(5.7)
Contents
|hx, ei i|2 .
P∞
i=1
λi ei ∈ H, we have
2 " *
+#2
∞
∞
X
X
kxk2 λi ei − Re x,
λi ei
≤ r2 kxk2
i=1
i=1
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and since
2
∞
∞
X
X
λi ei =
|λi |2 ,
* i=1
+ i=1
∞
∞
X
X
λi ei =
Re λi hx, ei i ,
Re x,
i=1
i=1
hence, by (5.7) we deduce
kxk
2
∞
X
i=1
"
2
*
|λi | − Re x,
∞
X
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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+#2
λi ei
≤ r2 kxk2 ,
giving the first inequality in (5.4).
The second inequality is obvious by the modulus property.
The last inequality follows by the Cauchy-Bunyakovsky-Schwarz inequality
∞
2
∞
∞
X
X
X
2
λi hx, ei i ≤
|λi |
|hx, ei i|2 .
i=1
S.S. Dragomir
i=1
i=1
i=1
by the last inequality in (5.4) on subtracting in both
The inequality (5.5)
Pfollows
∞
sides the quantity i=1 |hx, ei i|2 < ∞.
The following result provides a generalization for the reverse of Bessel’s
inequality obtained in [12].
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Theorem 5.2. Let (H; h·, ·i) and (ei )i∈N be as in Theorem 5.1. Suppose that
Γ = (Γi )i∈N ∈ `2 (K) , γ = (γi )i∈N ∈ `2 (K) are sequences of real or complex
numbers such that
∞
X
(5.8)
Re (Γi γi ) > 0.
i=1
If x ∈ H is such that either
(5.9)
∞
X
Γi + γi 1
ei ≤
x −
2
2
i=1
∞
X
! 12
2
|Γi − γi |
i=1
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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S.S. Dragomir
or, equivalently,
(5.10)
Re
*∞
X
Γi ei − x, x −
i=1
∞
X
Title Page
+
γi ei
≥0
holds, then we have the inequalities
(5.11)
1
kxk ≤ ·
4
2
1
≤ ·
4
≤
1
·
4
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i=1
2
Re Γi + γi hx, ei i
P∞
i=1 Re (Γi γi )
P∞
2
i=1 Γi + γi hx, ei i
P∞
i=1 Re (Γi γi )
P∞
∞
|Γi + γi |2 X
Pi=1
|hx, ei i|2 .
∞
Re
(Γ
γ
)
i i i=1
i=1
P∞
i=1
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The constant 14 is best possible in all inequalities in (5.11).
We also have the inequalities:
P∞
∞
∞
2 X
X
1
2
2
i=1 |Γi − γi |
P
(5.12)
0 ≤ kxk −
|hx, ei i| ≤ · ∞
|hx, ei i|2 .
4
i=1 Re (Γi γi ) i=1
i=1
Here the constant
1
4
is also best possible.
Proof. Since Γ, γ ∈ `2 (K) , then also 21 (Γ ± γ) ∈ `2 (K) , showing that the
series
∞ ∞
∞ X
X
Γi − γi 2
Γi + γi 2 X
,
and
Re (Γi γi )
2 2 i=1
i=1
i=1
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
S.S. Dragomir
are convergent. Also, the series
∞
X
i=1
Γi ei ,
∞
X
i=1
γi ei and
∞
X
γi + Γi
i=1
2
Title Page
ei
are convergent in the Hilbert space H.
The equivalence of the conditions (5.9) and (5.10) follows by the fact that
in an inner product
space
we1 have, for x, z, Z ∈ H, Re hZ − x, x − zi ≥ 0 is
≤ kZ − zk , and we omit the details.
equivalent to x − z+Z
2
2
Now, we observe that the inequalities (5.11) and (5.12) follow from Theorem
1
P∞
2 2
1
i
5.1 on choosing λi = γi +Γ
,
i
∈
N
and
r
=
|Γ
−
γ
|
.
i
i
i=1
2
2
1
The fact that 4 is the best constant in both (5.11) and (5.12) follows from
Theorem 2.2 and Corollary 2.3, and we omit the details.
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Remark 5.1. Note that (5.11) improves (1.17) and (5.12) improves (1.18), that
have been obtained in [12].
For some recent results related to Bessel inequality, see [1], [5], [14], and
[16].
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
S.S. Dragomir
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6.
Some Grüss Type Inequalities for Orthonormal
Families
The following result related to Grüss inequality in inner product spaces, holds.
Theorem 6.1. Let (H; h·, ·i) be an infinite dimensional Hilbert space over the
real or complex number field K, and (ei )i∈N an orthornormal family in H. Assume that λ = (λi )i∈N , µ = (µi )i∈N ∈ `2 (K) and r1 , r2 > 0 with the properties that
∞
X
(6.1)
|λi |2 > r12 ,
i=1
If x, y ∈ H are such that
∞
X
(6.2)
λi ei ≤ r1 ,
x −
i=1
∞
X
|µi |2 > r22 .
Reverses of Schwarz, Triangle
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S.S. Dragomir
i=1
Title Page
∞
X
µi ei ≤ r2 ,
y −
i=1
then we have the inequalities
∞
X
(6.3) hx, yi −
hx, ei i hei , yi
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i=1
r1 r2
qP
≤ qP
∞
2
∞
2
2
2
i=1 |λi | − r1
i=1 |µi | − r2
v
u∞
∞
X
uX
·t
|hx, ei i|2
|hy, ei i|2
i=1
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r1 r2 kxk kyk
qP
.
2
∞
2
2
2
|λ
|
−
r
|µ
|
−
r
i
i
1
2
i=1
i=1
≤ qP
∞
Proof.
P∞ Applying Schwarz’s inequality for the vectors x −
i=1 hy, ei i ei , we have
P∞
i=1
hx, ei i ei , y −
*
+2
∞
∞
X
X
hx, ei i ei , y −
hy, ei i ei x−
i=1
i=1
2 2
∞
∞
X
X
≤ x −
hx, ei i ei y −
hy, ei i ei .
(6.4)
i=1
S.S. Dragomir
i=1
Since
Title Page
*
x−
∞
X
hx, ei i ei , y −
i=1
and
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
∞
X
+
hy, ei i ei
= hx, yi −
i=1
∞
X
hx, ei i hei , yi
i=1
2
∞
∞
X
X
2
hx, ei i ei = kxk −
|hx, ei i|2 ,
x −
i=1
i=1
hence, by (5.5) applied for x and y, and from (6.4), we deduce the first part of
(6.3).
The second part follows by Bessel’s inequality.
The following Grüss type inequality may be stated as well.
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Theorem 6.2. Let (H; h·, ·i) be an infinite dimensional Hilbert space and (ei )i∈N
an orthornormal family in H. Suppose that (Γi )i∈N , (γi )i∈N , (φi )i∈N , (Φi )i∈N ∈
`2 (K) are sequences of real and complex numbers such that
∞
X
(6.5)
Re (Γi γi ) > 0,
i=1
∞
X
Re Φi φi > 0.
i=1
If x, y ∈ H are such that either
(6.6)
∞
1
X
Γ
+
γ
i
i
· ei ≤
x −
2
2
i=1
∞
1
X
Φ
+
φ
i
i
· ei ≤
y −
2
2
i=1
∞
X
! 12
|Γi − γi |2
S.S. Dragomir
i=1
∞
X
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
! 12
|Φi − φi |2
Title Page
i=1
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or, equivalently,
(6.7)
Re
Re
*∞
X
* i=1
∞
X
i=1
Γi ei − x, x −
∞
X
+
≥ 0,
γi ei
i=1
Φi ei − y, y −
∞
X
i=1
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+
φi ei
≥ 0,
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holds, then we have the inequality
∞
X
hx, ei i hei , yi
(6.8)
hx, yi −
i=1
1
1
P∞
2 2
2 2 P∞
1
i=1 |Γi − γi |
i=1 |Φi − φi |
≤ · P
4 ( ∞ Re (Γ γ )) 12 P∞ Re Φ φ 12
i i
i i
i=1
i=1
1
! 12
!
∞
∞
2
X
X
×
|hx, ei i|2
|hy, ei i|2
i=1
i=1
1
2 2
|Γ
−
γ
|
|Φ
−
φ
|
1
i
i
i
i
i=1
kxk kyk .
≤ · Pi=1
4 [ ∞ Re (Γ γ )] 12 P∞ Re Φ φ 12
i i
i i
i=1
i=1
P∞
The constant
1
4
1
2 2
P∞
is best possible in the first inequality.
Proof. Follows by (5.12) and (6.4).
The best constant follows from Theorem 4.2, and we omit the details.
Remark 6.1. We note that the inequality (6.8) is better than the inequality (3.3)
in [12]. We omit the details.
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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S.S. Dragomir
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7.
Integral Inequalities
Let (Ω, Σ, µ) be a measurable space consisting of a set Ω, a σ−algebra of
parts Σ and a countably additive and positive measure µ on Σ with
R values in
R∪ {∞} . Let ρ ≥ 0 be a Lebesgue measurable function on Ω with Ω ρ (s) dµ (s) =
1. Denote by L2ρ (Ω, K) the Hilbert space of all real or complex valued functions
defined on Ω and 2 − ρ−integrable on Ω, i.e.,
Z
(7.1)
ρ (s) |f (s)|2 dµ (s) < ∞.
Ω
It is obvious that the following inner product
Z
(7.2)
hf, giρ :=
ρ (s) f (s) g (s)dµ (s) ,
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
S.S. Dragomir
Ω
1
generates the norm kf kρ := Ω ρ (s) |f (s)| dµ (s) 2 of L2ρ (Ω, K) , and all the
above results may be stated for integrals.
It is important to observe that, if
h
i
(7.3)
Re f (s) g (s) ≥ 0 for µ − a.e. s ∈ Ω,
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2
R
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then, obviously,
Close
Z
(7.4)
ρ (s) f (s) g (s)dµ (s)
Re hf, giρ = Re
Ω
Z
h
i
=
ρ (s) Re f (s) g (s) dµ (s) ≥ 0.
Ω
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The reverse is evidently not true in general.
Moreover, if the space is real, i.e., K = R, then a sufficient condition for
(7.4) to hold is:
(7.5)
f (s) ≥ 0, g (s) ≥ 0 for µ − a.e. s ∈ Ω.
We provide now, by the use of certain results obtained in Section 2, some
integral inequalities that may be used in practical applications.
Proposition 7.1. Let f, g ∈ L2ρ (Ω, K) and r > 0 with the properties that
(7.6)
|f (s) − g (s)| ≤ r ≤ |g (s)| for µ − a.e. s ∈ Ω.
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
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S.S. Dragomir
Then we have the inequalities
Z
Z
2
(7.7)
ρ (s) |f (s)| dµ (s) ρ (s) |g (s)|2 dµ (s)
0≤
Ω
Ω
2
Z
− ρ (s) f (s) g (s)dµ (s)
Ω
Z
Z
2
≤
ρ (s) |f (s)| dµ (s) ρ (s) |g (s)|2 dµ (s)
Ω
Ω
2
Z
−
ρ (s) Re f (s) g (s) dµ (s)
Ω
Z
2
≤r
ρ (s) |g (s)|2 dµ (s) .
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Ω
The constant 1 in front of r2 is best possible.
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The proof follows by Theorem 2.1 and we omit the details.
Proposition 7.2. Let f, g ∈ L2ρ (Ω, K) and γ, Γ ∈ K such that Re (Γγ) > 0 and
h
i
(7.8)
Re (Γg (s) − f (s)) f (s) − γg (s) ≥ 0 for µ − a.e. s ∈ Ω.
Then we have the inequalities
Z
Z
2
(7.9)
ρ (s) |f (s)| dµ (s) ρ (s) |g (s)|2 dµ (s)
Ω
Ω
io2
n h
R
Γ
+
γ
ρ
(s)
f
(s)
g
(s)dµ
(s)
Re
Ω
1
≤ ·
Re (Γγ)
4
2
2 Z
1 |Γ + γ| ρ (s) f (s) g (s)dµ (s) .
≤ ·
4 Re (Γγ)
Ω
The constant
1
4
is best possible in both inequalities.
The proof follows by Theorem 2.2 and we omit the details.
Corollary 7.3. With the assumptions of Proposition 7.2, we have the inequality
Z
Z
2
(7.10)
0≤
ρ (s) |f (s)| dµ (s) ρ (s) |g (s)|2 dµ (s)
Ω
Ω
Z
2
− ρ (s) f (s) g (s)dµ (s)
Ω
2
2 Z
1 |Γ − γ| .
≤ ·
ρ
(s)
f
(s)
g
(s)dµ
(s)
4 Re (Γγ) Ω
The constant
1
4
is best possible.
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
S.S. Dragomir
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Remark 7.1. If the space is real and we assume, for M ≥ m > 0, that
mg (s) ≤ f (s) ≤ M g (s) , for µ − a.e. s ∈ Ω,
(7.11)
then, by (7.9) and (7.10), we deduce the inequalities
Z
2
Z
ρ (s) [g (s)]2 dµ (s)
Ω
Z
2
1 (M + m)2
ρ (s) f (s) g (s) dµ (s) .
≤ ·
4
mM
Ω
ρ (s) [f (s)] dµ (s)
(7.12)
Ω
and
S.S. Dragomir
Z
(7.13)
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
Z
ρ (s) [f (s)]2 dµ (s) ρ (s) [g (s)]2 dµ (s)
Ω
Ω
Z
2
−
ρ (s) f (s) g (s) dµ (s)
Ω
Z
2
1 (M − m)2
≤ ·
ρ (s) f (s) g (s) dµ (s) .
4
mM
Ω
0≤
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The inequality (7.12) is known in the literature as Cassel’s inequality.
The following Grüss type integral inequality for real or complex-valued
functions also holds.
R
Proposition 7.4. Let f, g, h ∈ L2ρ (Ω, K) with Ω ρ (s) |h (s)|2 dµ (s) = 1 and
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a, A, b, B ∈ K such that Re (Aa) , Re Bb > 0 and
h
i
Re (Ah (s) − f (s)) f (s) − ah (s) ≥ 0,
h
i
Re (Bh (s) − g (s)) g (s) − bh (s) ≥ 0,
for µ−a.e. s ∈ Ω. Then we have the inequalities
Z
ρ (s) f (s) g (s)dµ (s)
Ω
Z
Z
− ρ (s) f (s) h (s)dµ (s) ρ (s) h (s) g (s)dµ (s)
Ω
≤
The constant
1
4
S.S. Dragomir
Ω
|A − a| |B − b|
1
·q
4
Re (Aa) Re Bb
Z
Z
× ρ (s) f (s) h (s)dµ (s) ρ (s) h (s) g (s)dµ (s)
Ω
Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
Ω
is best possible.
The proof follows by Theorem 4.2.
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Remark 7.2. All the other inequalities in Sections 3 – 6 may be used in a similar
way to obtain the corresponding integral inequalities. We omit the details.
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Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
S.S. Dragomir
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Reverses of Schwarz, Triangle
and Bessel Inequalities in Inner
Product Spaces
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