Journal of Inequalities in Pure and
Applied Mathematics
A POTPOURRI OF SCHWARZ RELATED INEQUALITIES IN
INNER PRODUCT SPACES (II)
volume 7, issue 1, article 14,
2006.
S.S. DRAGOMIR
School of Computer Science and Mathematics
Victoria University
PO Box 14428, Melbourne City
Victoria 8001, Australia.
EMail: sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/dragomir
Received 28 June, 2005;
accepted 17 September, 2005.
Communicated by: A. Lupaş
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
196-05
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Abstract
Further inequalities related to the Schwarz inequality in real or complex inner
product spaces are given.
2000 Mathematics Subject Classification: 46C05, 26D15.
Key words: Schwarz inequality, Inner product spaces, Reverse inequalities.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Quadratic Schwarz Related Inequalities . . . . . . . . . . . . . . . . . . 6
3
Other Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4
Applications for the Triangle Inequality . . . . . . . . . . . . . . . . . . 22
References
A Potpourri of Schwarz Related
Inequalities in Inner Product
Spaces (II)
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, that may be written in two forms:
|hx, yi|2 ≤ kxk2 kyk2 ,
(1.1)
x, y ∈ H
(quadratic form)
x, y ∈ H
(simple form).
or, equivalently,
|hx, yi| ≤ kxk kyk ,
(1.2)
A Potpourri of Schwarz Related
Inequalities in Inner Product
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The case of equality holds in (1.1) or in (1.2) if and only if the vectors x and y
are linearly dependent.
In the previous paper [6], several results related to Schwarz inequalities have
been established. We recall few of them below:
1. If x, y ∈ H\ {0} and kxk ≥ kyk , then

r−1
1 2 kxk

kx − yk2 if r ≥ 1

 2 r kyk
(1.3)
kxk kyk − Re hx, yi ≤



1
2
kxk
kyk
1−r
kx − yk2
if r < 1.
2. If (H; h·, ·i) is complex, α ∈ C with Re α, Im α > 0 and x, y ∈ H are
such that
Im
α
x −
≤r
(1.4)
·
y
Re α S.S. Dragomir
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then
(1.5)
kxk kyk − Re hx, yi ≤
1 Re α 2
·
r .
2 Im α
3. If α ∈ K\ {0} , then for any x, y ∈ H
2
α
(1.6)
kxk kyk − Re
hx, yi
|α|2
1 [|Re α| kx − yk + |Im α| kx + yk]2
≤ ·
.
2
|α|2
4. If p ≥ 1, then for any x, y ∈ H one has
 1

(kxk + kyk)2p − kx + yk2p p ,

1
(1.7) kxk kyk − Re hx, yi ≤ ×
2 
1
 kx − yk2p − |kxk − kyk|2p p .
5. If α, γ > 0 and β ∈ K with |β|2 ≥ αγ then for x, a ∈ H with a 6= 0 and
12
2
|β|
−
αγ
β
x − a ≤
(1.8)
kak ,
α α
one has
(1.9)
A Potpourri of Schwarz Related
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S.S. Dragomir
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Re β Re hx, ai + Im β Im hx, ai
√
αγ
|β| |hx, ai|
≤ √
αγ
kxk kak ≤
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and
(1.10)
kxk2 kak2 − |hx, ai|2 ≤
|β|2 − αγ
|hx, ai|2 .
αγ
The aim of this paper is to provide other results related to the Schwarz inequality. Applications for reversing the generalised triangle inequality are also
given.
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S.S. Dragomir
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2.
Quadratic Schwarz Related Inequalities
The following result holds.
Theorem 2.1. Let (H; h·, ·i) be a complex inner product space and x, y ∈ H,
α ∈ [0, 1] . Then
(2.1)
α kty − xk2 + (1 − α) kity − xk2 kyk2
≥ kxk2 kyk2 − [(1 − α) Im hx, yi + α Re hx, yi]2 ≥ 0
A Potpourri of Schwarz Related
Inequalities in Inner Product
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for any t ∈ R.
Proof. Firstly, recall that for a quadratic polynomial P : R → R, P (t) =
at2 + 2bt + c, a > 0, we have that
b
ac − b2
(2.2)
inf P (t) = P −
=
.
t∈R
a
a
Now, consider the polynomial P : R → R given by
(2.3)
2
2
P (t) := α kty − xk + (1 − α) kity − xk .
Since
kty − xk2 = t2 kyk2 − 2t Re hx, yi + kxk2
and
kity − xk2 = t2 kyk2 − 2t Im hx, yi + kxk2 ,
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hence
P (t) = t2 kyk2 − 2t [α Re hx, yi + (1 − α) Im hx, yi] + kxk2 .
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By the definition of P (see (2.3)), we observe that P (t) ≥ 0 for every t ∈ R,
therefore 14 ∆ ≤ 0, i.e.,
[(1 − α) Im hx, yi + α Re hx, yi]2 − kxk2 kyk2 ≤ 0,
proving the second inequality in (2.1).
The first inequality follows by (2.2) and the theorem is proved.
The following particular cases are of interest.
A Potpourri of Schwarz Related
Inequalities in Inner Product
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Corollary 2.2. For any x, y ∈ H one has the inequalities:
(2.4)
kty − xk2 kyk2 ≥ kαk2 kyk2 − [Re hx, yi]2 ≥ 0;
(2.5)
kity − xk2 kyk2 ≥ kαk2 kyk2 − [Im hx, yi]2 ≥ 0;
S.S. Dragomir
and
(2.6)
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1
kty − xk2 + kity − xk2 kyk2
2
≥ kxk2 kyk2 −
Im hx, yi + Re hx, yi
2
2
≥ 0,
Close
The following corollary may be stated as well:
Corollary 2.3. Let x, y ∈ H and Mi , mi ∈ R, i ∈ {1, 2} such that Mi ≥ mi >
0, i ∈ {1, 2} . If either
Re hM1 y − x, x − m1 yi ≥ 0 and
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(2.7)
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Re hM2 iy − x, x − im2 yi ≥ 0,
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or, equivalently,
(2.8)
x − M1 + m1 y ≤
2
M
+
m
2
2
x −
iy ≤
2
1
(M1 − m1 ) kyk
2
1
(M2 − m2 ) kyk
2
and
hold, then
(2.9)
(0 ≤) kxk2 kyk2 − [(1 − α) Im hx, yi + α Re hx, yi]2
1
≤ kyk4 α (M1 − m1 )2 + (1 − α) (M2 − m2 )2
4
A Potpourri of Schwarz Related
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S.S. Dragomir
for any α ∈ [0, 1] .
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Proof. It is easy to see that, if x, z, Z ∈ H, then the following statements are
equivalent:
(i) Re hZ − x, x − zi ≥ 0,
1
≤ kZ − zk .
(ii) x − z+Z
2
2
Utilising this property one may simply realize that the statements (2.7) and (2.8)
are equivalent.
Now, on making use of (2.8) and (2.1), one may deduce the desired inequality (2.9).
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Remark 1. If one assumes that M1 = M2 = M, m1 = m2 = m in either (2.7)
or (2.8), then
(2.10)
(0 ≤) kxk2 kyk2 − [(1 − α) Im hx, yi + α Re hx, yi]2
1
≤ kyk4 (M − m)2
4
for each α ∈ [0, 1] .
Remark 2. Corollary 2.3 may be seen as a potential source of some reverse
results for the Schwarz inequality. For instance, if x, y ∈ H and M ≥ m > 0
are such that either
1
M +m (2.11) Re hM y − x, x − myi ≥ 0 or x −
y
≤ (M − m) kyk
2
2
hold, then
(2.12)
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(0 ≤) kxk2 kyk2 − [Re hx, yi]2 ≤
1
(M − m)2 kyk4 .
4
If x, y ∈ H and N ≥ n > 0 are such that either
N +n 1
(2.13) Re hN iy − x, x − niyi ≥ 0 or x −
iy ≤ (N − n) kyk
2
2
hold, then
(2.14)
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(0 ≤) kxk2 kyk2 − [Im hx, yi]2 ≤
1
(N − n)2 kyk4 .
4
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We notice that (2.12) is an improvement of the inequality
(0 ≤) kxk2 kyk2 − |hx, yi|2 ≤
1
(M − m)2 kyk4
4
that has been established in [4] under the same condition (2.11) given above.
The following result may be stated as well.
Theorem 2.4. Let (H; h·, ·i) be a real or complex inner product space and
x, y ∈ H, α ∈ [0, 1] . Then
(2.15) α kty − xk2 + (1 − α) ky − txk2 α kyk2 + (1 − α) kxk2
≥ (1 − α) kxk2 + α kyk2 α kxk2 + (1 − α) kyk2
A Potpourri of Schwarz Related
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− [Re hx, yi]2 ≥ 0
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for any t ∈ R.
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Proof. Consider the polynomial P : R → R given by
(2.16)
2
2
P (t) := α kty − xk + (1 − α) ky − txk .
Since
kty − xk2 = t2 kyk2 − 2t Re hx, yi + kxk2
and
ky − txk2 = t2 kxk2 − 2t Re hx, yi + kyk2 ,
hence
P (t) = α kyk2 + (1 − α) kxk2 t2 − 2t Re hx, yi + α kxk2 + (1 − α) kyk2
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for any t ∈ R.
By the definition of P (see (2.16)), we observe that P (t) ≥ 0 for every
t ∈ R, therefore 41 ∆ ≤ 0, i.e., the second inequality in (2.15) holds true.
The first inequality follows by (2.2) and the theorem is proved.
Remark 3. We observe that if either α = 0 or α = 1, then (2.15) will generate
the same reverse of the Schwarz inequality as (2.4) does.
Corollary 2.5. If x, y ∈ H, then
2
(2.17)
2
2
kty − xk + ky − txk kxk + kyk
·
2
2
≥
A Potpourri of Schwarz Related
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2
kxk2 + kyk2
2
S.S. Dragomir
!2
2
− [Re hx, yi] ≥ 0
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for any t ∈ R and
(2.18)
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kx ± yk2 α kyk2 + (1 − α) kxk2
≥ (1 − α) kxk2 + α kyk2 α kxk2 + (1 − α) kyk2
− [Re hx, yi]2 ≥ 0
kxk2 + kyk2
2
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for any α ∈ [0, 1] .
In particular, we have
(2.19) kx ± yk2 ·
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2
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− [Re hx, yi]2 ≥ 0.
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In [7, p. 210], C.S. Lin has proved the following reverse of the Schwarz
inequality in real or complex inner product spaces (H; h·, ·i) :
(0 ≤) kxk2 kyk2 − |hx, yi|2 ≤
1
kxk2 kx − ryk2
r2
for any r ∈ R, r 6= 0 and x, y ∈ H.
The following slightly more general result may be stated:
Theorem 2.6. Let (H; h·, ·i) be a real or complex inner product space. Then
for any x, y ∈ H and α ∈ K (C, R) with α 6= 0 we have
(2.20)
(0 ≤) kxk2 kyk2 − |hx, yi|2 ≤
1
2
2
2 kxk kx − αyk .
|α|
The case of equality holds in (2.20) if and only if
Re hx, αyi = kxk .
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Proof. Observe that
I (α) := kxk2 kx − αyk2 − |α|2 kxk2 kyk2 − |hx, yi|2
= kxk2 kxk2 − 2 Re [ᾱ hx, yi] + |α|2 kyk2
− |α|2 kxk2 kyk2 + |α|2 |hx, yi|2
4
2
2
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2
= kxk − 2 kxk Re [ᾱ hx, yi] + |α| |hx, yi| .
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Since
(2.22)
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2
(2.21)
A Potpourri of Schwarz Related
Inequalities in Inner Product
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Re [ᾱ hx, yi] ≤ |α| |hx, yi| ,
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hence
(2.23)
I (α) ≥ kxk4 − 2 kxk2 |α| |hx, yi| + |α|2 |hx, yi|2
2
= kxk2 − |α| |hx, yi| ≥ 0.
Conversely, if (2.21) holds true, then I (α) = 0, showing that the equality case
holds in (2.20).
Now, if the equality case holds in (2.20), then we must have equality in (2.22)
and in (2.23) implying that
Re [hx, αyi] = |α| |hx, yi|
and
2
|α| |hx, yi| = kxk
A Potpourri of Schwarz Related
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which imply (2.21).
The following result may be stated.
Corollary 2.7. Let (H; h·, ·i) be as above and x, y ∈ H, α ∈ K\ {0} and r > 0
such that |α| ≥ r. If
(2.24)
kx − αyk ≤ r kyk ,
then
q
|α|2 − r2
(2.25)
|α|
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≤
|hx, yi|
(≤ 1) .
kxk kyk
Proof. From (2.24) and (2.20) we have
r2
2
2
kxk2 kyk2 − |hx, yi|2 ≤
2 kxk kyk ,
|α|
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that is,
|α|2 − r2
2
|α|
kxk2 kyk2 ≤ |hx, yi|2 ,
which is clearly equivalent to (2.25).
Remark 4. Since for Γ, γ ∈ K the following statements are equivalent
(i) Re hΓy − x, x − γyi ≥ 0,
(ii) x − γ+Γ
· y ≤ 12 |Γ − γ| kyk ,
2
A Potpourri of Schwarz Related
Inequalities in Inner Product
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hence by Corollary 2.7 we deduce
S.S. Dragomir
1
2
(2.26)
2 [Re (Γγ̄)]
|hx, yi|
≤
,
|Γ + γ|
kxk kyk
provided Re (Γγ̄) > 0, an inequality that has been obtained in a different way
in [3].
Corollary 2.8. If x, y ∈ H, α ∈ K\ {0} and ρ > 0 such that kx − αyk ≤ ρ,
then
(2.27)
ρ2
2
2
2
2
(0 ≤) kxk kyk − |hx, yi| ≤
2 kxk .
|α|
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3.
Other Inequalities
The following result holds.
Proposition 3.1. Let x, y ∈ H \ {0} and ε ∈ (0, 12 ]. If
(3.1)
(0 ≤) 1 − ε −
√
1 − 2ε ≤
√
kxk
≤ 1 − ε + 1 − 2ε,
kyk
then
(3.2)
(0 ≤) kxk kyk − Re hx, yi ≤ ε kx − yk2 .
Proof. If x = y, then (3.2) is trivial.
Suppose x 6= y. Utilising the inequality (2.5) from [6], we can state that
kxk kyk − Re hx, yi
2 kxk kyk
≤
2
kx − yk
(kxk + kyk)2
for any x, y ∈ H \ {0} , x 6= y.
Now, if we assume that
2 kxk kyk
≤ ε,
(kxk + kyk)2
then, after some manipulation, we get that
ε kxk2 + 2 (ε − 1) kxk kyk + ε kyk2 ≥ 0,
which, for ε ∈ (0, 21 ] and y 6= 0, is clearly equivalent to (3.1).
The proof is complete.
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The following result may be stated:
Proposition 3.2. Let (H; h·, ·i) be a real or complex inner product space. Then
for any x, y ∈ H and ϕ ∈ R one has:
kxk kyk − [cos 2ϕ · Re hx, yi + sin 2ϕ · Im hx, yi]
1
≤ [|cos ϕ| kx − yk + |sin ϕ| kx + yk]2 .
2
Proof. For ϕ ∈ R, consider the complex number α = cos ϕ − i sin ϕ. Then
α2 = cos 2ϕ − i sin 2ϕ, |α| = 1 and by the inequality (1.6) we deduce the
desired result.
A Potpourri of Schwarz Related
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From a different perspective, we may consider the following results as well:
Theorem 3.3. Let (H; h·, ·i) be a real or complex inner product space, α ∈ K
with |α − 1| = 1. Then for any e ∈ H with kek = 1 and x, y ∈ H, we have
(3.3)
|hx, yi − α hx, ei he, yi| ≤ kxk kyk .
The equality holds in (3.3) if and only if there exists a λ ∈ K such that
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(3.4)
α hx, ei e = x + λy.
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Proof. It is known that for u, v ∈ H, we have equality in the Schwarz inequality
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|hu, vi| ≤ kuk kvk
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(3.5)
if and only if there exists a λ ∈ K such that u = λv.
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If we apply (3.5) for u = α hx, ei e − x, v = y, we get
(3.6)
|hα hx, ei e − x, yi| ≤ kα hx, ei e − xk kyk
with equality iff there exists a λ ∈ K such that
α hx, ei e = x + λy.
Since
2
2
2
2
2
kα hx, ei e − xk = |α| |hx, ei| − 2 Re [α] |hx, ei| + kxk
= |α|2 − 2 Re [α] |hx, ei|2 + kxk2
= |α − 1|2 − 1 |hx, ei|2 + kxk2
= kxk2
A Potpourri of Schwarz Related
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and
hα hx, ei e − x, yi = α hx, ei he, yi − hx, yi
hence by (3.6) we deduce the desired inequality (3.3).
Remark 5. If α = 0 in (3.3), then it reduces to the Schwarz inequality.
Remark 6. If α 6= 0, then (3.3) is equivalent to
1
hx, ei he, yi − hx, yi ≤ 1 kxk kyk .
(3.7)
|α|
α
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Utilising the continuity property of modulus for complex numbers, i.e.,
|z − w| ≥ ||z| − |w|| we then obtain
|hx, ei he, yi| − 1 |hx, yi| ≤ 1 kxk kyk ,
|α|
|α|
which implies that
|hx, ei he, yi| ≤
(3.8)
For e =
(3.9)
z
,
kzk
1
[|hx, yi| + kxk kyk] .
|α|
z 6= 0, we get from (3.8) that
S.S. Dragomir
|hx, zi hz, yi| ≤
1
[|hx, yi| + kxk kyk] kzk2
|α|
for any α ∈ K\ {0} with |α − 1| = 1 and x, y, z ∈ H.
For α = 2, we get from (3.9) the Buzano inequality [1]
(3.10)
|hx, zi hz, yi| ≤
1
[|hx, yi| + kxk kyk] kzk2
2
for any x, y, z ∈ H.
Remark 7. In the case of real spaces the condition |α − 1| = 1 is equivalent to
either α = 0 or α = 2. For α = 2 we deduce from (3.7) that
(3.11)
A Potpourri of Schwarz Related
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1
1
[hx, yi − kxk kyk] ≤ hx, ei he, yi ≤ [hx, yi + kxk kyk]
2
2
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for any x, y ∈ H and e ∈ H with kek = 1, which implies Richard’s inequality
[8]:
(3.12)
1
1
[hx, yi − kxk kyk] kzk2 ≤ hx, zi hz, yi ≤ [hx, yi + kxk kyk] kzk2 ,
2
2
for any x, y, z ∈ H.
The following result concerning a generalisation for orthornormal families
of the inequality (3.3) may be stated.
Theorem 3.4. Let (H; h·, ·i) be a real or complex inner product space, {ei }i∈F
a finite orthonormal family, i.e., hei , ej i = δij for i, j ∈ F , where δij is Kronecker’s delta and αi ∈ K, i ∈ F such that |αi − 1| = 1 for each i ∈ F.
Then
X
(3.13)
αi hx, ei i hei , yi ≤ kxk kyk .
hx, yi −
i∈F
The equality holds in (3.13) if and only if there exists a constant λ ∈ K such
that
X
(3.14)
αi hx, ei i ei = x + λy.
S.S. Dragomir
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i∈F
Proof. As above, by Schwarz’s inequality, we have
*
+ X
X
(3.15)
α
hx,
e
i
e
−
x,
y
≤
α
hx,
e
i
e
−
x
kyk
i
i
i
i
i
i
i∈F
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Inequalities in Inner Product
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i∈F
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P
with equality if and only if there exists a λ ∈ K such that i∈F αi hx, ei i ei =
x + λy.
Since
2
X
αi hx, ei i ei − x
i∈F
2
*
+ X
X
2
αi hx, ei i ei = kxk − 2 Re x,
αi hx, ei i ei + i∈F
i∈F
X
X
= kxk2 − 2
αi hx, ei i hx, ei i +
|αi |2 |hx, ei i|2
i∈F
i∈F
2
= kxk +
X
2
|hx, ei i|
2
|αi | − 2 Re αi
S.S. Dragomir
i∈F
= kxk2 +
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A Potpourri of Schwarz Related
Inequalities in Inner Product
Spaces (II)
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|hx, ei i|2 |αi − 1|2 − 1
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i∈F
= kxk2 ,
hence the inequality (3.13) is obtained.
Remark 8. If the space is real, then the nontrivial case one can get from (3.13)
is for all αi = 2, obtaining the inequality
(3.16)
X
1
1
[hx, yi − kxk kyk] ≤
hx, ei i hei , yi ≤ [hx, yi + kxk kyk]
2
2
i∈F
that has been obtained in [5].
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Corollary 3.5. With the above assumptions, we have
X
X
αi hx, ei i hei , yi ≤ |hx, yi| + hx, yi −
αi hx, ei i hei , yi
(3.17)
i∈F
i∈F
≤ |hx, yi| + kxk kyk ,
x, y ∈ H,
where |αi − 1| = 1 for each i ∈ F and {ei }i∈F is an orthonormal family in H.
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4.
Applications for the Triangle Inequality
In 1966, Diaz and Metcalf [2] proved the following reverse of the triangle
inequality:
n
n
X X
(4.1)
x
≥
r
kxi k ,
i
i=1
i=1
provided the vectors xi in the inner product space (H; h·, ·i) over the real or
complex number field K are nonzero and
(4.2)
0≤r≤
Re hxi , ai
kxi k
for each i ∈ {1, . . . , n} ,
S.S. Dragomir
where a ∈ H, kak = 1. The equality holds in (4.2) if and only if
!
n
n
X
X
(4.3)
xi = r
kxi k a.
i=1
Proposition 4.1. Let e ∈ H with kek = 1, ε ∈ 0, 12 and xi ∈ H, i ∈
{1, . . . , n} with the property that
√
√
(4.4)
(0 ≤) 1 − ε − 1 − 2ε ≤ kxi k ≤ 1 − ε + 1 − 2ε
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for each i ∈ {1, . . . , n} . Then
n
n
n
X
X
X
kxi k ≤ xi + ε
kxi − ek2 .
i=1
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i=1
The following result may be stated:
(4.5)
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i=1
i=1
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Proof. Utilising Proposition 3.1 for x = xi and y = e, we can state that
kxi k − Re hxi , ei ≤ ε kxi − ek2
for each i ∈ {1, . . . , n} . Summing over i from 1 to n, we deduce that
* n
+
n
n
X
X
X
(4.6)
kxi k ≤ Re
xi , e + ε
kxi − ek2 .
i=1
i=1
i=1
By Schwarz’s inequality in (H; h·, ·i) , we also have
* n
+ * n
+ * n
+
X
X
X
Re
xi , e ≤ Re
xi , e ≤ (4.7)
xi , e i=1
i=1
i=1
n
n
X
X
≤
xi kek = xi .
i=1
i=1
Therefore, by (4.6) and (4.7) we deduce (4.5).
In the same spirit, we can prove the following result as well:
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Proposition 4.2. Let (H; h·, ·i) be a real or complex inner product space and
e ∈ H with kek = 1. Then for any ϕ ∈ R one has the inequality:
n
n
n
X
1X
X
[|cos ϕ| kxi − ek + |sin ϕ| kxi + ek]2 .
(4.8)
kxi k ≤ xi +
2
i=1
i=1
i=1
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Proof. Applying Proposition 3.2 for x = xi and y = e, we have:
(4.9)
kxi k ≤ cos 2ϕ · Re hxi , ei + sin 2ϕ · Im hxi , ei
1
+ [|cos ϕ| kxi − ek + |sin ϕ| kxi + ek]2
2
for any i ∈ {1, . . . , n} .
Summing in (4.5) over i from 1 to n, we have:
* n
+
* n
+
n
X
X
X
(4.10)
kxi k ≤ cos 2ϕ · Re
xi , e + sin 2ϕ · Im
xi , e
i=1
i=1
+
n
1X
2
i=1
[|cos ϕ| kxi − ek + |sin ϕ| kxi + ek]2 .
we have
(4.11)
cos 2ϕ · Re
+
xi , e
+ sin 2ϕ · Im
i=1
2
S.S. Dragomir
i=1
Now, by the elementary inequality for the real numbers m, p, M and P,
1
1
mM + pP ≤ m2 + p2 2 M 2 + P 2 2 ,
* n
X
A Potpourri of Schwarz Related
Inequalities in Inner Product
Spaces (II)
* n
X
+
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21
≤ cos 2ϕ + sin 2ϕ
" *
+#2 " * n
+#2  12
n
X
X

×  Re
xi , e
+ Im
xi , e
i=1
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i=1
2
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*
+ n
n
n
X
X X
=
xi , e ≤ xi kek = xi ,
i=1
i=1
i=1
where for the last inequality we used Schwarz’s inequality in (H; h·, ·i) .
Finally, by (4.10) and (4.11) we deduce the desired result (4.8).
A Potpourri of Schwarz Related
Inequalities in Inner Product
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References
[1] M.L. BUZANO, Generalizzazione della disiguaglianza di Cauchy-Schwarz
(Italian), Rend. Sem. Mat. Univ. e Politech. Torino, 31 (1971/73), 405-409
(1974).
[2] J.B. DIAZ AND F.T. METCALF, A complementary inequality in Hilbert
and Banach spaces, Proc. Amer. Math. Soc., 17(1) (1966), 88-97.
[3] S.S. DRAGOMIR, Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces, J. Inequal. Pure & Appl. Math., 5(3)
(2004), Art. 76. [ONLINE http://jipam.vu.edu.au/article.
php?sid=432].
[4] S.S. DRAGOMIR, A counterpart of Schwarz’s inequality in inner product
spaces, East Asian Math. J., 20(1) (2004), 1-10.
[5] S.S. DRAGOMIR, Inequalities for orthornormal families of vectors in inner product spaces related to Buzano’s, Richard’s and Kurepa’s results,
RGMIA Res. Rep. Coll., 7(2004), Article 25. [ONLINE http://rgmia.
vu.edu.au/v7(E).html].
A Potpourri of Schwarz Related
Inequalities in Inner Product
Spaces (II)
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[6] S.S. DRAGOMIR, A potpourri of Schwarz related inequalities in inner
product spaces (I), J. Inequal. Pure & Appl. Math., 6(3) (2004), Art.
59. [ONLINE http://jipam.vu.edu.au/article.php?sid=
532].
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[7] C.S. LIN, Cauchy-Schwarz inequality and generalized Bernstein-type inequalities, in Inequality Theory and Applications, Vol. 1, Eds. Y.J. Cho,
J.K. Kim and S.S. Dragomir, Nova Science Publishers Inc., N.Y., 2001.
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[8] U. RICHARD, Sur des inégalités du type Wirtinger et leurs application aux
équationes différentielles ordinaires, Colloquium of Analysis held in Rio de
Janeiro, August, 1972, pp. 233-244.
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