Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 2, Article 42, 2003
SOME GRÜSS TYPE INEQUALITIES IN INNER PRODUCT SPACES
S.S. DRAGOMIR
S CHOOL OF C OMPUTER S CIENCE & M ATHEMATICS
V ICTORIA U NIVERSITY,
PO B OX 14428, MCMC
M ELBOURNE , V ICTORIA 8001,
AUSTRALIA .
sever@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au.SSDragomirWeb.html
Received 12 March, 2003; accepted 12 April, 2003
Communicated by B. Mond
A BSTRACT. Some new Grüss type inequalities in inner product spaces and applications for
integrals are given.
Key words and phrases: Grüss’ Inequality, Inner products, Integral inequalities, Discrete Inequalities.
2000 Mathematics Subject Classification. Primary 26D15, 46D05 Secondary 46C99.
1. I NTRODUCTION
In [1], the author has proved the following Grüss type inequality in real or complex inner
product spaces.
Theorem 1.1. Let (H, h·, ·i) be an inner product space over K (K = R,C) and e ∈ H, kek = 1.
If ϕ, γ, Φ, Γ are real or complex numbers and x, y are vectors in H such that the conditions
Re hΦe − x, x − ϕei ≥ 0 and Re hΓe − y, y − γei ≥ 0
(1.1)
hold, then we have the inequality
The constant
1
|Φ − ϕ| · |Γ − γ| .
4
is best possible in the sense that it cannot be replaced by a smaller constant.
|hx, yi − hx, ei he, yi| ≤
(1.2)
1
4
Some particular cases of interest for integrable functions with real or complex values and the
corresponding discrete versions are listed below.
Corollary 1.2. Let f, g : [a, b] → K (K = R,C) be Lebesgue integrable and such that
h
i
h
i
(1.3)
Re (Φ − f (x)) f (x) − ϕ ≥ 0, Re (Γ − g (x)) g (x) − γ ≥ 0
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
032-03
2
S.S. D RAGOMIR
for a.e. x ∈ [a, b] , where ϕ, γ, Φ, Γ are real or complex numbers and z̄ denotes the complex
conjugate of z. Then we have the inequality
Z b
Z b
Z b
1
1
1
(1.4) f (x) g (x)dx −
f (x) dx ·
g (x)dx
b−a a
b−a a
b−a a
1
≤ |Φ − ϕ| · |Γ − γ| .
4
1
The constant 4 is best possible.
The discrete case is embodied in
Corollary 1.3. Let x, y ∈Kn and ϕ, γ, Φ, Γ are real or complex numbers such that
Re [(Φ − xi ) (xi − ϕ)] ≥ 0,
(1.5)
Re [(Γ − yi ) (yi − γ)] ≥ 0
for each i ∈ {1, . . . , n} . Then we have the inequality
n
n
n
1 X
1 X 1
1X
xi ·
yi ≤ |Φ − ϕ| · |Γ − γ| .
xi yi −
(1.6)
n
n i=1
n i=1 4
i=1
The constant
1
4
is best possible.
For other applications of Theorem 1.1, see the recent paper [2].
In the present paper we show that the condition (1.1) may be replaced by an equivalent but
simpler assumption and a new proof of Theorem 1.1 is produced. A refinement of the Grüss
type inequality (1.2) , some companions and applications for integrals are pointed out as well.
2. A N E QUIVALENT A SSUMPTION
The following lemma holds.
Lemma 2.1. Let a, x, A be vectors in the inner product space (H, h·, ·i) over K (K = R,C) with
a 6= A. Then
Re hA − x, x − ai ≥ 0
if and only if
1
a
+
A
x −
≤ kA − ak .
2 2
Proof. Define
2
1
a
+
A
2
.
I1 := Re hA − x, x − ai , I2 := kA − ak − x
−
4
2 A simple calculation shows that
I1 = I2 = Re [hx, ai + hA, xi] − Re hA, ai − kxk2
and thus, obviously, I1 ≥ 0 iff I2 ≥ 0, showing the required equivalence.
The following corollary is obvious
Corollary 2.2. Let x, e ∈ H with kek = 1 and δ, ∆ ∈ K with δ 6= ∆. Then
Re h∆e − x, x − δei ≥ 0
if and only if
x − δ + ∆ · e ≤ 1 |∆ − δ| .
2
2
J. Inequal. Pure and Appl. Math., 4(2) Art. 42, 2003
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G RÜSS ’ T YPE I NEQUALITIES
3
Remark 2.3. If H = C, then
Re [(A − x) (x̄ − ā)] ≥ 0
if and only if
1
a
+
A
x −
≤ |A − a| ,
2 2
where a, x, A ∈ C. If H = R, and A > a then a ≤ x ≤ A if and only if x −
a+A 2
≤ 12 |A − a| .
The following lemma also holds.
Lemma 2.4. Let x, e ∈ H with kek = 1. Then one has the following representation
0 ≤ kxk2 − |hx, ei|2 = inf kx − λek2 .
(2.1)
λ∈K
Proof. Observe, for any λ ∈ K, that
hx − λe, x − hx, ei ei = kxk2 − |hx, ei|2 − λ he, xi − he, xi kek2
= kxk2 − |hx, ei|2 .
Using Schwarz’s inequality, we have
2
kxk2 − |hx, ei|2 = |hx − λe, x − hx, ei ei|2
≤ kx − λek2 kx − hx, ei ek2
= kx − λek2 kxk2 − |hx, ei|2
giving the bound
kxk2 − |hx, ei|2 ≤ kx − λek2 , λ ∈ K.
(2.2)
Taking the infimum in (2.2) over λ ∈ K, we deduce
kxk2 − |hx, ei|2 ≤ inf kx − λek2 .
λ∈K
Since, for λ0 = hx, ei , we get kx − λ0 ek2 = kxk2 − |hx, ei|2 , then the representation (2.1) is
proved.
We are able now to provide a different proof for the Grüss type inequality in inner product
spaces mentioned in the Introduction, than the one from paper [1].
Theorem 2.5. Let (H, h·, ·i) be an inner product space over K (K = R,C) and e ∈ H, kek = 1.
If ϕ, γ, Φ, Γ are real or complex numbers and x, y are vectors in H such that the conditions (1.1)
hold, or, equivalently, the following assumptions
ϕ+Φ 1
γ+Γ 1
(2.3)
x − 2 · e ≤ 2 |Φ − ϕ| ,
y − 2 · e ≤ 2 |Γ − γ|
are valid, then one has the inequality
|hx, yi − hx, ei he, yi| ≤
(2.4)
The constant
1
4
1
|Φ − ϕ| · |Γ − γ| .
4
is best possible.
Proof. It can be easily shown (see for example the proof of Theorem 1 from [1]) that
1 1
(2.5)
|hx, yi − hx, ei he, yi| ≤ kxk2 − |hx, ei|2 2 kyk2 − |hy, ei|2 2 ,
J. Inequal. Pure and Appl. Math., 4(2) Art. 42, 2003
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S.S. D RAGOMIR
for any x, y ∈ H and e ∈ H, kek = 1. Using Lemma 2.4 and the conditions (2.3) we obviously
have that
1
1
ϕ
+
Φ
2
2 2
≤ |Φ − ϕ|
kxk − |hx, ei|
= inf kx − λek ≤ x
−
·
e
2
λ∈K
2
and
1
1
γ
+
Γ
2
2 2
≤ |Γ − γ|
kyk − |hy, ei|
= inf ky − λek ≤ y
−
·
e
2
λ∈K
2
and by (2.5) the desired inequality (2.4) is obtained.
The fact that 14 is the best possible constant, has been shown in [1] and we omit the details.
3. A R EFINEMENT
OF THE
G RÜSS I NEQUALITY
The following result improving (1.1) holds
Theorem 3.1. Let (H, h·, ·i) be an inner product space over K (K = R,C) and e ∈ H, kek = 1.
If ϕ, γ, Φ, Γ are real or complex numbers and x, y are vectors in H such that the conditions
(1.1), or, equivalently, (2.3) hold, then we have the inequality
|hx, yi − hx, ei he, yi|
1
1
1
≤ |Φ − ϕ| · |Γ − γ| − [Re hΦe − x, x − ϕei] 2 [Re hΓe − y, y − γei] 2 .
4
Proof. As in [1], we have
(3.2)
|hx, yi − hx, ei he, yi|2 ≤ kxk2 − |hx, ei|2 kyk2 − |hy, ei|2 ,
(3.1)
(3.3)
2
2
h
kxk − |hx, ei| = Re (Φ − hx, ei) hx, ei − ϕ
i
− Re hΦe − x, x − ϕei
and
(3.4)
h
i
kyk2 − |hy, ei|2 = Re (Γ − hy, ei) hy, ei − γ − Re hΓe − x, x − γei .
Using the elementary inequality
4 Re ab ≤ |a + b|2 ; a, b ∈ K (K = R,C)
we may state that
(3.5)
h
i 1
Re (Φ − hx, ei) hx, ei − ϕ ≤ |Φ − ϕ|2
4
and
i 1
h
Re (Γ − hy, ei) hy, ei − γ ≤ |Γ − γ|2 .
4
Consequently, by (3.2) − (3.6) we may state that
1 2
1
2
2
2
(3.7) |hx, yi − hx, ei he, yi| ≤
|Φ − ϕ| − [Re hΦe − x, x − ϕei]
4
1 2
1
2
2
×
|Γ − γ| − [Re hΓe − y, y − γei]
.
4
Finally, using the elementary inequality for positive real numbers
m2 − n2 p2 − q 2 ≤ (mp − nq)2
(3.6)
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G RÜSS ’ T YPE I NEQUALITIES
5
we have
1 2
1
2
|Φ − ϕ| − [Re hΦe − x, x − ϕei] 2
4
1 2
1
2
×
|Γ − γ| − [Re hΓe − y, y − γei] 2
4
2
1
1
1
≤
|Φ − ϕ| · |Γ − γ| − [Re hΦe − x, x − ϕei] 2 [Re hΓe − y, y − γei] 2 ,
4
giving the desired inequality (3.1) .
4. S OME C OMPANION I NEQUALITIES
The following companion of the Grüss inequality in inner product spaces holds.
Theorem 4.1. Let (H, h·, ·i) be an inner product space over K (K = R,C) and e ∈ H, kek = 1.
If γ, Γ ∈ K and x, y ∈ H are such that
x+y x+y
(4.1)
Re Γe −
,
− γe ≥ 0
2
2
or, equivalently,
x + y γ + Γ 1
2 − 2 · e ≤ 2 |Γ − γ| ,
(4.2)
then we have the inequality
The constant
1
|Γ − γ|2 .
4
is best possible in the sense that it cannot be replaced by a smaller constant.
Re [hx, yi − hx, ei he, yi] ≤
(4.3)
1
4
Proof. Start with the well known inequality
1
(4.4)
Re hz, ui ≤ kz + uk2 ; z, u ∈ H.
4
Since
hx, yi − hx, ei he, yi = hx − hx, ei e, y − hy, ei ei
then using (4.4) we may write
(4.5)
Re [hx, yi − hx, ei he, yi] = Re [hx − hx, ei e, y − hy, ei ei]
1
≤ kx − hx, ei e + y − hy, ei ek2
4
2
x + y
x+y
=
−
, e · e
2
2
2
x + y 2 x + y
.
=
−
,
e
2 2
If we apply Grüss’ inequality in inner product spaces for, say, a = b =
2
x + y 2 x + y
≤ 1 |Γ − γ|2 .
(4.6)
−
,
e
2 2
4
x+y
,
2
we get
Making use of (4.5) and (4.6) we deduce (4.3) .
J. Inequal. Pure and Appl. Math., 4(2) Art. 42, 2003
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6
S.S. D RAGOMIR
The fact that 14 is the best possible constant in (4.3) follows by the fact that if in (4.1) we
choose x = y, then it becomes Re hΓe − x, x − γei ≥ 0, implying 0 ≤ kxk2 − |hx, ei|2 ≤
1
|Γ − γ|2 , for which, by Grüss’ inequality in inner product spaces, we know that the constant
4
1
is best possible.
4
The following corollary might be of interest if one wanted to evaluate the absolute value of
Re [hx, yi − hx, ei he, yi] .
Corollary 4.2. Let (H, h·, ·i) be an inner product space over K (K = R,C) and e ∈ H, kek =
1. If γ, Γ ∈ K and x, y ∈ H are such that
x±y x±y
(4.7)
Re Γe −
,
− γe ≥ 0
2
2
or, equivalently,
x ± y γ + Γ 1
2 − 2 · e ≤ 2 |Γ − γ| ,
(4.8)
then we have the inequality
1
|Γ − γ|2 .
4
If the inner product space H is real, then (for m, M ∈ R, M > m)
x±y x±y
,
− me ≥ 0
(4.10)
Me −
2
2
(4.9)
|Re [hx, yi − hx, ei he, yi]| ≤
or, equivalently,
(4.11)
x ± y m + M 1
· e
≤ 2 (M − m) ,
2 −
2
implies
(4.12)
|hx, yi − hx, ei he, yi| ≤
In both inequalities (4.9) and (4.12) , the constant
1
4
1
(M − m)2 .
4
is best possible.
Proof. We only remark that, if
x−y x−y
Re Γe −
,
− γe ≥ 0
2
2
holds, then by Theorem 4.1, we get
Re [− hx, yi + hx, ei he, yi] ≤
1
|Γ − γ|2 ,
4
showing that
1
Re [hx, yi − hx, ei he, yi] ≥ − |Γ − γ|2 .
4
Making use of (4.3) and (4.13) we deduce the desired result (4.9) .
(4.13)
Finally, we may state and prove the following dual result as well
J. Inequal. Pure and Appl. Math., 4(2) Art. 42, 2003
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G RÜSS ’ T YPE I NEQUALITIES
7
Proposition 4.3. Let (H, h·, ·i) be an inner product space over K (K = R,C) and e ∈ H, kek =
1. If ϕ, Φ ∈ K and x, y ∈ H are such that
h
i
(4.14)
Re (Φ − hx, ei) hx, ei − ϕ ≤ 0,
then we have the inequalities
(4.15)
1
kx − hx, ei ek ≤ [Re hx − Φe, x − ϕei] 2
√
1
2
≤
kx − Φek2 + kx − ϕek2 2 .
2
Proof. We know that the following identity holds true (see (3.3))
h
i
(4.16)
kxk2 − |hx, ei|2 = Re (Φ − hx, ei) hx, ei − ϕ + Re hx − Φe, x − ϕei .
Using the assumption (4.14) and the fact that
kxk2 − |hx, ei|2 = kx − hx, ei ek2 ,
by (4.16) we deduce the first inequality in (4.15) .
The second inequality in (4.15) follows by the fact that for any v, w ∈ H one has
Re hw, vi ≤
1
kwk2 + kvk2 .
2
The proposition is thus proved.
5. I NTEGRAL I NEQUALITIES
Let (Ω, Σ, µ) be a measure space consisting of a set Ω, a σ−algebra of parts Σ and a countably
additive and positive measure µ on Σ with values in R∪ {∞} . Denote by L2 (Ω, K) the Hilbert
space of all real or complex valued functions f defined on Ω and 2−integrable on Ω, i.e.,
Z
|f (s)|2 dµ (s) < ∞.
Ω
The following proposition holds
R
Proposition 5.1. If f, g, h ∈ L2 (Ω, K) and ϕ, Φ, γ, Γ ∈ K, are such that Ω |h (s)|2 dµ (s) = 1
and
Z
i
h
(5.1)
Re (Φh (s) − f (s)) f (s) − ϕh (s) dµ (s) ≥ 0,
Ω
Z
h
i
Re (Γh (s) − g (s)) g (s) − γh (s) dµ (s) ≥ 0
Ω
or, equivalently
(5.2)
! 12
2
Z Φ
+
ϕ
1
f (s) −
h (s) dµ (s)
≤ |Φ − ϕ| ,
2
2
Ω
! 12
2
Z Γ
+
γ
1
g (s) −
h (s) dµ (s)
≤ |Γ − γ| ,
2
2
Ω
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8
S.S. D RAGOMIR
then we have the following refinement of the Grüss integral inequality
(5.3)
Z
Z
Z
f (s) g (s)dµ (s) − f (s) h (s)dµ (s) h (s) g (s)dµ (s)
Ω
Ω
Ω
Z
h
i
1
Re (Φh (s) − f (s)) f (s) − ϕh (s) dµ (s)
≤ |Φ − ϕ| · |Γ − γ| −
4
Ω
12
Z
h
i
×
Re (Γh (s) − g (s)) g (s) − γh (s) dµ (s) .
Ω
The constant
1
4
is best possible.
The proof follows by Theorem 3.1 on choosing H = L2 (Ω, K) with the inner product
Z
hf, gi :=
f (s) g (s)dµ (s) .
Ω
We omit the details.
Remark 5.2. It is obvious that a sufficient condition for (5.1) to hold is
h
i
Re (Φh (s) − f (s)) f (s) − ϕh (s) ≥ 0,
and
h
i
Re (Γh (s) − g (s)) g (s) − γh (s) ≥ 0,
for µ−a.e. s ∈ Ω, or equivalently,
Φ
+
ϕ
f (s) −
≤
h
(s)
2
g (s) − Γ + γ h (s) ≤
2
1
|Φ − ϕ| |h (s)| and
2
1
|Γ − γ| |h (s)| ,
2
for µ−a.e. s ∈ Ω.
The following result may be stated as well.
Corollary 5.3. If z, Z, t, T ∈ K, µ (Ω) < ∞ and f, g ∈ L2 (Ω, K) are such that:
i
h
(5.4)
Re (Z − f (s)) f (s) − z̄ ≥ 0,
h
i
Re (T − g (s)) g (s) − t̄ ≥ 0 for a.e. s ∈ Ω
or, equivalently
(5.5)
z
+
Z
f (s) −
≤
2 g (s) − t + T ≤
2 J. Inequal. Pure and Appl. Math., 4(2) Art. 42, 2003
1
|Z − z| ,
2
1
|T − t| for a.e. s ∈ Ω
2
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G RÜSS ’ T YPE I NEQUALITIES
9
then we have the inequality
Z
Z
Z
1
1
1
g (s)dµ (s)
(5.6) f (s) g (s)dµ (s) −
f (s) dµ (s) ·
µ (Ω) Ω
µ (Ω) Ω
µ (Ω) Ω
Z
h
i
1
1
Re (Z − f (s)) f (s) − z̄ dµ (s)
≤ |Z − z| |T − t| −
4
µ (Ω) Ω
12
Z
h
i
×
Re (T − g (s)) g (s) − t̄ dµ (s) .
Ω
Using Theorem 4.1 we may state the following result as well.
R
Proposition 5.4. If f, g, h ∈ L2 (Ω, K) and γ, Γ ∈ K are such that Ω |h (s)|2 dµ (s) = 1 and
(
#)
"
Z
f (s) + g (s)
f (s) + g (s)
·
− γ̄ h̄ (s)
dµ (s) ≥ 0
(5.7)
Re Γh (s) −
2
2
Ω
or, equivalently,
! 12
2
Z f (s) + g (s) γ + Γ
1
−
h (s) dµ (s)
≤ |Γ − γ| ,
2
2
2
Ω
(5.8)
then we have the inequality
Z
h
i
(5.9)
I :=
Re f (s) g (s) dµ (s)
Ω
Z
Z
− Re
f (s) h (s)dµ (s) · h (s) g (s)dµ (s)
Ω
Ω
1
|Γ − γ|2 .
4
If (5.7) and (5.8) hold with “ ± ” instead of “ + ”, then
1
(5.10)
|I| ≤ |Γ − γ|2 .
4
Remark 5.5. It is obvious that a sufficient condition for (5.7) to hold is
(
#)
"
f (s) + g (s)
f (s) + g (s)
(5.11)
Re Γh (s) −
·
− γ̄ h̄ (s)
≥0
2
2
≤
for a.e. s ∈ Ω, or equivalently
f (s) + g (s) γ + Γ
1
≤ |Γ − γ| |h (s)| for a.e. s ∈ Ω.
(5.12)
−
h
(s)
2
2
2
Finally, the following corollary holds.
Corollary 5.6. If Z, z ∈ K, µ (Ω) < ∞ and f, g ∈ L2 (Ω, K) are such that
"
!#
f (s) + g (s)
f (s) + g (s)
(5.13)
Re Z −
−z
≥ 0 for a.e. s ∈ Ω
2
2
or, equivalently
(5.14)
f (s) + g (s) z + Z 1
≤ |Z − z| for a.e. s ∈ Ω,
−
2
2 2
J. Inequal. Pure and Appl. Math., 4(2) Art. 42, 2003
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10
S.S. D RAGOMIR
then we have the inequality
Z
h
i
1
J :=
Re f (s) g (s) dµ (s)
µ (Ω) Ω
Z
Z
1
1
− Re
f (s) dµ (s) ·
g (s)dµ (s)
µ (Ω) Ω
µ (Ω) Ω
1
≤ |Z − z|2 .
4
If (5.13) and (5.14) hold with “ ± ” instead of “ + ”, then
1
(5.15)
|J| ≤ |Z − z|2 .
4
Remark 5.7. It is obvious that if one chooses the discrete measure above, then all the inequalities in this section may be written for sequences of real or complex numbers. We omit the
details.
R EFERENCES
[1] S.S. DRAGOMIR, A generalization of Grüss’ inequality in inner product spaces and applications, J.
Math. Anal. Appl., 237 (1999), 74–82.
[2] S.S. DRAGOMIR AND I. GOMM, Some integral and discrete versions of the Grüss inequality for
real and complex functions and sequences, RGMIA Res. Rep. Coll., 5(3) (2003), Article 9, ONLINE
[http://rgmia.vu.edu.au/v5n3.html]
J. Inequal. Pure and Appl. Math., 4(2) Art. 42, 2003
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