Ann. Funct. Anal. 5 (2014), no. 1, 77–93
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL:www.emis.de/journals/AFA/
NEW GRÜSS TYPE INEQUALITIES FOR
RIEMANN–STIELTJES INTEGRAL WITH MONOTONIC
INTEGRATORS AND APPLICATIONS
MOHAMMAD W. ALOMARI1∗ AND SEVER S. DRAGOMIR2
Dedicated to Professor Tsuyoshi Ando with affection
Communicated by T. Riedel
Abstract. In this paper several new inequalities of Grüss’ type for Riemann–
Stieltjes integral with monotonic nondecreasing integrators under various assumptions for integrands are proved. Applications for functions of selfadjoint
operators on complex Hilbert spaces are provided as well.
1. Introduction
The Čebyšev functional defined by
Z b
Z b
Z b
1
1
1
T (f, g) =
f (t) g (t) dt −
f (t) dt ·
g (t) dt
(1.1)
b−a a
b−a a
b−a a
has interesting applications in the approximation of the integral of a product as
pointed out in the references below.
The problem of bounding the Čebyšev functional has a long history, starting
with Grüss [21] inequality in 1935, where he had proved that for two integrable
functions f, g such that φ ≤ f (x) ≤ Φ and γ ≤ f (x) ≤ Γ for any x ∈ [a, b], the
inequality
1
|C (f, g)| ≤ (Φ − φ) (Γ − γ)
(1.2)
4
holds, and the constant 41 is the best possible.
Date: Received: 25 June 2013; Accepted: 22 September 2013.
∗
Corresponding author.
2000 Mathematics Subject Classification. Primary 26D10; Secondary 26D15, 47A63.
Key words and phrases. Grüss inequality, Functions of bounded variation, Hölder continuous
functions, Riemann–Stieltjes integral, Selfadjoint operators, Functions of selfadjoint operators.
77
78
M.W. ALOMARI, S.S. DRAGOMIR
After that many authors have studied the functional (1.1) and therefore, several
bounds under various assumptions for the functions involved have been obtained.
For some new results and generalizations the reader may refer to [1]–[20], [23]
and the references therein.
One of the recent generalization of (1.1) was considered by Dragomir in [9].
Namely, he has introduced the following Čebyšev functional for the Riemann–
Stieltjes integral
Z b
1
T (f, g; u) :=
f (t) g (t) du (t)
(1.3)
u (b) − u (a) a
Z b
Z b
1
1
−
f (t) du (t) ·
g (t) du (t)
u (b) − u (a) a
u (b) − u (a) a
under the assumptions that f, g are continuous on [a, b] and u is of bounded
variation on [a, b] with u(b) 6= u(a).
By simple computations with the Riemann–Stieltjes integral, Dragomir [9] established the identity:
Z b
f (a) + f (b)
1
T (f, g; u) =
f (t) −
u (b) − u (a) a
2
Z b
1
· g (t) −
g (s) du (s) du (t),
u (b) − u (a) a
(1.4)
to obtain several sharp bounds of the Čebyšev functional for the Riemann–
Stieltjes integral (1.3).
In this paper, some new Grüss’ type inequalities for the Riemann–Stieltjes
integral with monotonic nondecreasing integrators are proved. Applications for
functions of selfadjoint operators on complex Hilbert spaces via the spectral representation theorem are provided as well.
2. The Results
We may start with the following result:
Theorem 2.1. Let f : [a, b] → C be a p–Hf –Hölder continuous function on [a, b],
where p ∈ (0, 1] and Hf > 0 are given. Let g, u : [a, b] → R be such that g is
Riemann–Stieltjes integrable with respect to a monotonic non-decreasing function
u on [a, b] and there exists the real numbers γ, Γ such that γ ≤ g(x) ≤ Γ for all
x ∈ [a, b], then
|T (f, g; u)| ≤
1
2p+1
Hf (Γ − γ) (b − a)p .
(2.1)
NEW GRÜSS TYPE INEQUALITIES
79
Proof. Taking the modulus in (1.4) and utilizing the triangle inequality, we get
Z b
1
f (a) + f (b) |T (f, g; u)| ≤
f (t) −
(2.2)
u (b) − u (a) a 2
Z b
1
× g (t) −
g (s) du (s) du (t)
u (b) − u (a) a
1
f (a) + f (b) ≤
sup f (t) −
u (b) − u (a) t∈[a,b] 2
Z b
Z b
1
×
g (s) du (s) du (t).
g (t) − u (b) − u (a)
a
a
Now, using the same approach considered in [11], we define
2
Z b
Z b
1
1
I (g) :=
g (t) −
g (s) du (s) du (t).
u (b) − u (a) a
u (b) − u (a) a
Then, we have
Z b
Z b
1
1
2
I (g) =
g (t) − 2g (t)
g (s) du (s)
u (b) − u (a) a
u (b) − u (a) a
2 #
Z b
1
+
g (s) du (s)
du (t)
u (b) − u (a) a
2
Z b
Z b
1
1
2
=
g (t) du (t) −
g (s) du (s)
u (b) − u (a) a
u (b) − u (a) a
and
Z b
Z b
1
1
I (g) = Γ −
g (s) du (s)
g (s) du (s) − γ
u (b) − u (a) a
u (b) − u (a) a
Z b
1
−
(Γ − g (s)) (g (s) − γ) du (s).
u (b) − u (a) a
As γ ≤ g(t) ≤ Γ, for all t ∈ [a, b], then
Z b
(Γ − g (s)) (g (s) − γ) du (s) ≥ 0,
a
which implies
I (g)
≤ Γ−
1
u (b) − u (a)
Z
a
b
g (s) du (s)
1
u (b) − u (a)
Z
(2.3)
b
g (s) du (s) − γ
a
1
(Γ − γ)2 .
4
Using Cauchy–Buniakowski–Schwarz’s integral inequality we have
2
Z b
Z b
1
1
I (g) ≥
g (t) −
g (s) du (s) du (t)
u (b) − u (a) a u (b) − u (a) a
≤
80
M.W. ALOMARI, S.S. DRAGOMIR
and thus by (2.3) we get
Z b
Z b
1
1
g (t) −
du (t) ≤ 1 (Γ − γ) . (2.4)
g
(s)
du
(s)
u (b) − u (a) a
u (b) − u (a) a
2
Now, since f is of p–Hf –Hölder type on [a, b], then we have
f (t) − f (a) + f (b) = f (t) − f (a) + f (t) − f (b) 2
2
1
1
|f (t) − f (a)| + |f (t) − f (b)|
2
2
Hf
[(t − a)p + (b − t)p ] .
≤
2
≤
It follows that:
p
f
(a)
+
f
(b)
b
−
a
≤ Hf
sup f (t) −
.
2
2
(2.5)
t∈[a,b]
Combining (2.4) and (2.5) with (2.2), we get
1
f (a) + f (b) |T (f, g; u)| ≤
sup f (t) −
u (b) − u (a) t∈[a,b] 2
Z b
Z b
1
g (t) −
×
g (s) du (s) du (t)
u (b) − u (a) a
a
Hf
≤ p+1 (Γ − γ) (b − a)p ,
2
as required.
Corollary 2.2. Let g, u be as in Theorem 2.1. If f : [a, b] → C is Lf –Lipschitz
on [a, b], then
1
|T (f, g; u)| ≤ Lf (Γ − γ) (b − a) .
4
When the function f is of bounded variation we can state the following result
as well:
Theorem 2.3. Let g, u be as in Theorem 2.1. Let f : [a, b] → C be a function of
bounded variation on [a, b], then we have
b
_
1
|T (f, g; u)| ≤ (Γ − γ) (f ) .
4
a
The constant
1
4
(2.6)
is best possible.
Proof. As in Theorem 2.1, we observe that
|T (f, g; u)|
1
f (a) + f (b) ≤
sup f (t) −
u (b) − u (a) t∈[a,b] 2
Z b
Z b
1
×
g (s) du (s) du (t),
g (t) − u (b) − u (a)
a
a
(2.7)
NEW GRÜSS TYPE INEQUALITIES
81
and
1
u (b) − u (a)
Z b
Z b
1
1
g (t) −
g (s) du (s) du (t) ≤ (Γ − γ) . (2.8)
u (b) − u (a) a
2
a
Since f is of bounded variation on [a, b], then we have
f
(a)
+
f
(b)
sup f (t) −
2
t∈[a,b]
f (t) − f (a) + f (t) − f (b) = sup 2
t∈[a,b]
b
1
1_
≤ sup [|f (t) − f (a)| + |f (t) − f (b)|] ≤
(f ) ,
2 t∈[a,b]
2 a
(2.9)
for all t ∈ [a, b]. Finally, combining the inequalities (2.7)–(2.9), we obtain the
required result (2.6).
Assume that (2.6) holds with a constant C > 0,i.e.,
|T (f, g; u)| ≤ C (Γ − γ)
b
_
(f ) .
(2.10)
a
Consider the functions u (t) = t and f (t) = g (t) = sgn t −
Wb
a (f ) = 2, Γ − γ = 2 and
1
b−a
b
Z
Z
f (t) g (t) dt = 1,
a
1
4
Z
f (t) dt =
a
and from (2.10) we get C ≥
b
a+b
2
, t ∈ [a, b] . Then
b
g (t) dt = 0
a
which proves the sharpness of the constant.
We have the following result as well:
Theorem 2.4. Let g : [a, b] → C be such that g is of bounded variation on [a, b]
and u be a monotonic nondecreasing functions on [a, b], then we have
|T (f, g; u)| ≤

p Wb
 Hf (b − a) a (g) ,

1
2
Wb
a
(f )
Wb
a
(g) ,
where, Hf > 0 and p ∈ (0, 1] are given.
if f is Hf -p-Hölder
(2.11)
if f is of bounded variation
82
M.W. ALOMARI, S.S. DRAGOMIR
Proof. Using (1.4) we may write
|T (f, g; u)|
(2.12)
Z b
1
f
(a)
+
f
(b)
f (t) −
≤
u (b) − u (a) a 2
Z b
1
× g (t) −
g (s) du (s) du (t)
u (b) − u (a) a
Z b
1
f (a) + f (b) =
f (t) −
2
(u (b) − u (a))2 a Z b
×
[g (t) − g (s)] du (s) du (t)
a
1
(u (b) − u (a))2
Z
Z b f (a) + f (b) b
×
|g (t) − g (s)| du (s) du (t) .
f (t) −
2
≤
a
a
But since g is of bounded variation we then have
Z b
Z b
|g (t) − g (s)| du (s) ≤ sup |g (t) − g (s)|
du (s)
s∈[a,b]
a
≤
b
_
a
(g) [u (b) − u (a)] .
a
Therefore, if f is of p–Hölder type, then we have
|T (f, g; u)|
b
_
1
1
≤
(g)
2 u (b) − u (a) a
b
[|f (t) − f (a)| + |f (t) − f (b)|] du (t)
a
_
Hf
≤
(g)
2 (u (b) − u (a)) a
b
b
Z
_
Hf
(g)
=
2 (u (b) − u (a)) a
b
Z
[|t − a|p + |t − b|p ] du (t)
a
b
Z
[(t − a)p + (b − t)p ] du (t).
We have
Z b
Z b
Z b
p
p
p
[(t − a) + (b − t) ] du (t) =
(t − a) du (t) +
(b − t)p du (t) .
a
a
a
Utilising the integration by parts formula, we have
Z b
Z b
p
p
(t − a) du (t) = (b − a) u (b) − p
u (t) (t − a)p−1 dt
a
(2.13)
a
a
(2.14)
NEW GRÜSS TYPE INEQUALITIES
83
and
Z
b
p
Z
p
(b − t) du (t) = − (b − a) u (a) + p
a
b
u (t) (b − t)p−1 dt.
a
Now, since u is monotonic we have
Z b
1
u (t) (t − a)p−1 dt ≥ (b − a)p u (a) ,
p
a
and
Z
b
u (t) (b − t)p−1 dt ≤
a
1
(b − a)p u (b) .
p
Thus, by (2.14), we get
Z b
Z b
Z b
p
p
p
[(t − a) + (b − t) ] du (t) =
(t − a) du (t) +
(b − t)p du (t)
a
a
(2.15)
a
≤ 2 (b − a)p [u (b) − u (a)] ,
which gives by (2.13), that
b
_
Hf
|T (f, g; u)| ≤
(g)
2 (u (b) − u (a)) a
p
≤ Hf (b − a)
b
_
Z
b
[(t − a)p + (b − t)p ] du (t).
a
(g) ,
a
which prove the first part of inequality (2.11).
To prove the second part of (2.11), assume that f is of bounded variation, then
we have
|T (f, g; u)|
b
_
1
1
≤
(g)
2 u (b) − u (a) a
b
Z
b
[|f (t) − f (a)| + |f (t) − f (b)|] du (t)
a
b
_
1_
(g) (f )
≤
2 a
a
and thus the theorem is proved.
Theorem 2.5. Let g, u : [a, b] → R be such that g is of q-Hg –Hölder type on
[a, b], and u be a monotonic nondecreasing functions on [a, b], then we have

W
if f is of bounded variation
 Hg (b − a)q ba (f ) ,
|T (f, g; u)| ≤
(2.16)
 1
p+q
H H (b − a) , if f is Hf -p-Hölder
2p f g
where Hf > 0 and p ∈ (0, 1] are given.
84
M.W. ALOMARI, S.S. DRAGOMIR
Proof. Using (1.4), we may write
|T (f, g; u)|
(2.17)
Z b
1
f
(a)
+
f
(b)
f (t) −
≤
u (b) − u (a) a
2
Z b
1
× g (t) −
g (s) du (s) du (t)
u (b) − u (a) a
Z b
f
(a)
+
f
(b)
1
f (t) −
=
2
(u (b) − u (a))2 a Z b
× [g (t) − g (s)] du (s) du (t)
a
1
(u (b) − u (a))2
Z b
Z b f
(a)
+
f
(b)
f (t) −
×
|g (t) − g (s)| du (s) du (t)
2
a
a
1
f
(a)
+
f
(b)
f (t) −
≤
sup
2
(u (b) − u (a))2 t∈[a,b] Z b Z b
×
|g (t) − g (s)| du (s) du (t)
≤
a
a
Z b Z b
b
_
1
≤
(f )
|g (t) − g (s)| du (s) du (t).
2 (u (b) − u (a))2 a
a
a
Since g is of q-Hg –Holder type on [a, b], then we have
|T (f, g; u)|
Z b Z b
b
_
1
(f )
|g (t) − g (s)| du (s) du (t)
≤
2 (u (b) − u (a))2 a
a
a
Z b Z b
b
_
Hg
q
≤
(f )
|t − s| du (s) du (t).
2 (u (b) − u (a))2 a
a
a
(2.18)
NEW GRÜSS TYPE INEQUALITIES
85
Now, using the Riemann–Stieltjes integral and then utilizing the monotonicity of
u on [a, b] (twice), we get
Z b Z b
q
|t − s| du (s) du (t)
a
a
Z b Z t
Z b
q
q
=
(s − a) du (s) +
(b − s) du (s) du (t)
a
a
t
Z b
Z t
q
=
(t − a) u (t) − q
(s − a)q−1 u (s) ds
a
a
Z b
q
q−1
− (b − t) u (t) + q
(b − s) u (s) ds du (t)
t
Z
b
[(t − a)q u (t) − (t − a)q u (a) − (b − t)q u (t) + (b − t)q u (b)] du (t)
≤
a
Z
b
[(t − a)q u (b) − (t − a)q u (a) − (b − t)q u (a) + (b − t)q u (b)] du (t)
a
Z b
= [u (b) − u (a)]
[(t − a)q + (b − t)q ] du (t)
≤
a
q
≤ 2 (b − a) [u (b) − u (a)]2 ,
by (2.15).
which proves the first part of (2.16).
To prove the second part of (2.16), assume that f is of p-Hf –Hölder type, then
we have
|T (f, g; u)|
f (a) + f (b) 1
≤
sup f (t) −
2
(u (b) − u (a))2 t∈[a,b] Z b Z b
×
|g (t) − g (s)| du (s) du (t)
a
a
p Z b Z b
Hg
b−a
q
≤
Hf
|t − s| du (s) du (t)
2
2 (u (b) − u (a))2
a
a
p
Hg
b−a
=
2 (b − a)q (u (b) − u (a))2
2 Hf
2
2 (u (b) − u (a))
1
= p Hg Hf (b − a)p+q ,
2
which proves the second part of (2.16), and thus the proof is established.
3. Generalizations of some recent results
In this section we generalize some Grüss type inequalities for Lebesgue–Stieltjes
integral.
In the recent work [7], Dragomir has proved the following inequality:
86
M.W. ALOMARI, S.S. DRAGOMIR
Theorem 3.1. Let f : [a, b] → C be of bounded variation on [a, b] and g : [a, b] →
C a Lebesgue integrable function on [a, b], then
Z b
Z b
b
1_
1
1
|T (f, g)| ≤
(f )
g (t) −
g (s) ds dt,
(3.1)
2 a
b−a a
b−a a
W
where, ba (f ) denotes the total variation of f on the interval [a, b]. The constant
1
is best possible in (3.1).
2
Another result when both functions are of bounded variation, was considered
in the same paper [7], as follows:
Theorem 3.2. If f, g : [a, b] → C are of bounded variation on [a, b], then
b
b
_
1_
(f ) (g)
|T (f, g)| ≤
4 a
a
The constant
1
4
(3.2)
is best possible in (3.2).
The following result holds:
Theorem 3.3. Let f : [a, b] → C be of bounded variation on [a, b] and g : [a, b] →
C be such that g is a Lebesgue–Stieltjes integrable with respect to a monotonic
nondecreasing function u on [a, b], then
|T (f, g; u)|
(3.3)
Z b
Z b
b
1_
1
1
g (t) −
du (t).
≤
(f )
g
(s)
du
(s)
2 a
u (b) − u (a) a u (b) − u (a) a
The constant
1
2
is best possible.
Proof. Using (1.4) and utilizing the triangle inequality, we get
|T (f, g; u)|
Z b
1
f
(a)
+
f
(b)
f (t) −
≤
u (b) − u (a) a 2
Z b
1
× g (t) −
g (s) du (s) du (t)
u (b) − u (a) a
1
f
(a)
+
f
(b)
≤
sup f (t) −
u (b) − u (a) t∈[a,b]
2
Z b
Z b
1
du (t)
×
g
(t)
−
g
(s)
du
(s)
u (b) − u (a) a
a
Z b
Z b
b
1_
1
1
≤
(f )
g (t) −
g (s) du (s) du (t) .
2 a
u (b) − u (a) a
u (b) − u (a) a
Since f is of bounded variation on [a, b], then we have that
|f (t) − f (a)| + |f (t) − f (b)| ≤
b
_
a
(f )
NEW GRÜSS TYPE INEQUALITIES
87
for all t ∈ [a, b].
The sharpness
of (3.3) follows immediately by taking u (t) = t and f (t) =
a+b
sgn t − 2 and g (t) = t − a+b
. For the details see [7].
2
The variance of the function f : [a, b] → C which is square integrable on [a, b]
by D (f ) and is defined as [7]:
1/2
D (f ) := T f, f
"
2 #1/2
Z b
Z b
1
1
=
|f (t)|2 dt − f (t) dt
,
(3.4)
b−a a
b−a a
where f denotes the complex conjugate function of f .
A generalization of D (f ) in terms of Lebesgue–Stieltjes integral may be considered by defining the variance of the function f : [a, b] → C which is square
integrable with respect to a monotonic nondecreasing function u : [a, b] → C on
[a, b] by D (f ; u) and is defined as:
1/2
D (f ; u) := T f, f ; u
"
2 #1/2
Z b
Z b
1
1
=
|f (t)|2 du (t) − f (t) du (t)
.
u (b) − u (a) a
u (b) − u (a) a
(3.5)
Corollary 3.4. Let f : [a, b] → C be a function of bounded variation on [a, b],
then
b
1_
D (f ; u) ≤
(f ) .
(3.6)
2 a
Proof. Applying Theorem 3.3 for g = f we get
D2 (f ; u)
b
1_
1
≤
(f )
2 a
u (b) − u (a)
Z b
Z b
1
f (t) −
du (t).
f
(s)
du
(s)
u (b) − u (a) a
a
By the Cauchy–Bunyakovsky–Schwarz integral inequality we have
Z b
Z b
1
1
f (t) −
du (t) ≤ D (f ; u) .
f
(s)
du
(s)
u (b) − u (a)
u (b) − u (a)
a
(3.7)
(3.8)
a
Making use of (3.7) and (3.8) we get (3.6).
The sharpness follows by taking u (t) = t and f (t) = sgn t −
the details.
a+b
2
. We omit
Remark 3.5. If in Theorems 2.1–2.5, we set g = f , then several bounds for
D (f ; u) under various assumptions for the functions involved may be deduced.
The details are omitted.
In the following we obtain another result when both functions f, g are of
bounded variation, which therefore generalizes Theorem 3.2 and refines the second inequality of (2.11):
88
M.W. ALOMARI, S.S. DRAGOMIR
Theorem 3.6. Let f, g : [a, b] → C be two functions of bounded variation on
[a, b], then
b
b
_
1_
|T (f, g; u)| ≤
(f ) (g) .
(3.9)
4 a
a
The constant
1
4
is the best possible.
Proof. On making use of Theorem 3.3 and Corollary 3.4 we have
|T (f, g; u)|
(3.10)
Z b
Z b
b
1_
1
1
du (t)
≤
(f )
g
(t)
−
g
(s)
du
(s)
2 a
u (b) − u (a) a u (b) − u (a) a
b
b
b
_
1_
1_
≤
(f ) D (g; u) ≤
(f ) (g) .
2 a
4 a
a
The case of equality is obtained in (3.9) for f (t) = g (t) = sgn t −
[a, b].
a+b
2
, t ∈
Remark 3.7. By reconsidering the functions in Theorems 2.1–2.5 to be complex
valued functions and according to Remark 3.5, one may obtain several bounds
for T (f, g; u) using D (f ; u). The details are left to the interested readers.
4. Applications for selfadjoint operators
We denote by B (H) the Banach algebra of all bounded linear operators on
a complex Hilbert space (H; h·, ·i) . Let A ∈ B (H) be selfadjoint and let ϕλ be
defined for all λ ∈ R as follows

 1, for − ∞ < s ≤ λ,
ϕλ (s) :=
 0, for λ < s < +∞.
Then for every λ ∈ R the operator
Eλ := ϕλ (A)
(4.1)
is a projection which reduces A.
The properties of these projections are collected in the following fundamental
result concerning the spectral representation of bounded selfadjoint operators in
Hilbert spaces, see for instance [22, p. 256]:
Theorem 4.1 (Spectral Representation Theorem). Let A be a bonded selfadjoint
operator on the Hilbert space H and let m = min {λ |λ ∈ Sp (A)} =: min Sp (A)
and M = max {λ |λ ∈ Sp (A)} =: max Sp (A) . Then there exists a family of
projections {Eλ }λ∈R , called the spectral family of A, with the following properties
a) Eλ ≤ Eλ0 for λ ≤ λ0 ;
b) Em−0 = 0, EM = I and Eλ+0 = Eλ for all λ ∈ R;
NEW GRÜSS TYPE INEQUALITIES
89
c) We have the representation
Z
M
A=
λdEλ .
(4.2)
m−0
More generally, for every continuous complex-valued function ϕ defined on R
and for every ε > 0 there exists a δ > 0 such that
n
X
ϕ (λ0k ) Eλk − Eλk−1 ≤ ε
(4.3)
ϕ (A) −
k=1
whenever

λ0 < m = λ1 < · · · < λn−1 < λn = M,





λk − λk−1 ≤ δ for 1 ≤ k ≤ n,




 λ0 ∈ [λ , λ ] for 1 ≤ k ≤ n
k−1
k
k
this means that
Z
(4.4)
M
ϕ (A) =
ϕ (λ) dEλ ,
(4.5)
m−0
where the integral is of Riemann–Stieltjes type.
Corollary 4.2. With the assumptions of Theorem 4.1 for A, Eλ and ϕ we have
the representations
Z M
ϕ (A) x =
ϕ (λ) dEλ x for all x ∈ H
(4.6)
m−0
and
Z
M
hϕ (A) x, yi =
ϕ (λ) d hEλ x, yi for all x, y ∈ H.
(4.7)
ϕ (λ) d hEλ x, xi for all x ∈ H.
(4.8)
Moreover, we have the equality
Z M
2
kϕ (A) xk =
|ϕ (λ)|2 d kEλ xk2 for all x ∈ H.
(4.9)
m−0
In particular,
Z
M
hϕ (A) x, xi =
m−0
m−0
We observe that the function u (λ) := hEλ x, xi is monotonic nondecreasing and
right continuous on [m, M ] .
Theorem 4.3. Let A be a bonded selfadjoint operator on the Hilbert space H
and let m = min {λ |λ ∈ Sp (A)} =: min Sp (A) and M = max {λ |λ ∈ Sp (A)}
=: max Sp (A) .
(i) If f : [m, M ] → C is a p–Hf –Hölder continuous function on [m, M ], where
p ∈ (0, 1] and Hf > 0 are given and g : [m, M ] → R is continuous and
γ = mint∈[m,M ] g (t) , Γ = maxt∈[m,M ] g (t) , then
1
|C (f, g; A; x)| ≤ p+1 Hf (Γ − γ) (M − m)p ,
(4.10)
2
90
M.W. ALOMARI, S.S. DRAGOMIR
where C (f, g; A; x) denotes the following Čebyšev functional
C (f, g; A; x) := hf (A) g (A) x, xi − hf (A) x, xi · hg (A) x, xi
and x ∈ H with kxk = 1.
(ii) If f : [m, M ] → C is continuous and of bounded variation on [m, M ] while
g is as in (i), then
M
_
1
|C (f, g; A; x)| ≤ (Γ − γ) (f )
4
m
(4.11)
for any x ∈ H with kxk = 1.
(iii) If the function f is as in (i) and g : [m, M ] → C is continuous and of
bounded variation on [m, M ], then
|C (f, g; A; x)| ≤ Hf
M
_
(g) (M − m)p
(4.12)
m
for any x ∈ H with kxk = 1.
(iv) If f is of bounded variation and g is continuous, then
M
|C (f, g; A; x)| ≤
_
1
h|g (A) − hg (A) x, xi 1H | x, xi (f )
2
m
(4.13)
for any x ∈ H with kxk = 1. Moreover, if g is also of bounded variation,
then we have the simpler bound
M
_
1
|C (f, g; A; x)| ≤ h|g (A) − hg (A) x, xi 1H | x, xi (f )
2
m
M
≤
(4.14)
M
_
1_
(f ) (g)
4 m
m
for any x ∈ H with kxk = 1.
Proof. (i) Fix x ∈ H with kxk = 1. Let s > 0 and extend by continuity the
functions f and g to the interval [m − s, M ] by preserving their properties from
[m, M ] . Consider also the monotonic function u (λ) := hEλ x, xi which is monotonic nondecreasing and right continuous on [m − s, M ] .
Now, writing the inequality (2.1) for these functions we have
Z M
1
f (λ) g (λ) du (λ)
(4.15)
u (M ) − u (m − s)
m−s
Z M
1
−
f (λ) du (λ)
(4.16)
u (M ) − u (m − s) m−s
Z M
1
×
g (λ) du (λ)
u (M ) − u (m − s) m−s
1
≤ p+1 Hf (Γ − γ) (M − m + s)p .
2
NEW GRÜSS TYPE INEQUALITIES
91
Now, by letting s → 0+ and utilizing the representation (4.8) and the fact that
u (M ) = 1, u (m − 0) = 0 we get
|hf (A) g (A) x, xi − hf (A) x, xi hg (A) x, xi| ≤
1
2p+1
Hf (Γ − γ) (M − m)p
for x ∈ H with kxk = 1.
This proves the inequality (4.10).
The statement (ii) follows from (2.6), (iii) follows from (2.11), (iv) follows from
(3.3) and (3.10) by employing a similar argument.
The details are left to the interested reader.
The above results may provide some interesting inequalities for fundamental
functions of operators such as the power and logarithmic functions.
If we consider the function f : [0, ∞) → R, f (t) = tp with p ∈ (0, 1) then we
observe that this functions is p-Hölder continuous with the constant Hf = 1 on
any subinterval from [0, ∞). So, if A is a positive operator on the Hilbert space
H with the spectrum Sp (A) ⊂ [m, M ] ⊂ [0, ∞) then from (4.10) we have the
inequality
1
(Γ − γ) (M − m)p
(4.17)
2p+1
with x ∈ H with kxk = 1 and for any function g : [m, M ] → R that is continuous
and γ = mint∈[m,M ] g (t) , Γ = maxt∈[m,M ] g (t) .
Now, if in (4.17) we choose g (t) = tq with q > 0, then we get the inequality
p+q
A x, x − hAp x, xi hAq x, xi ≤ 1 (M q −mq ) (M − m)p
(4.18)
2p+1
for any x ∈ H with kxk = 1.
If m > 0, then by choosing g (t) = ln t we get the logarithmic inequality
1
M
p
p
|hA ln Ax, xi − hA x, xi hln Ax, xi| ≤ p+1 ln
(M − m)p
(4.19)
2
m
|hAp g (A) x, xi − hAp x, xi hg (A) x, xi| ≤
for any x ∈ H with kxk = 1.
Now, if we take the function g (t) = tq with q > 0 and if A is a positive operator
on the Hilbert space H with the spectrum Sp (A) ⊂ [m, M ] ⊂ [0, ∞), then we
get from (4.11)
M
|hf (A) Aq x, xi − hf (A) x, xi hAq x, xi| ≤
_
1
(M q −mq ) (f )
4
m
(4.20)
for any x ∈ H with kxk = 1, where f : [m, M ] → C is continuous and of bounded
variation on [m, M ].
If we take in (4.20) f (t) = tp with p > 0, then we get
p+q
A x, x − hAp x, xi hAq x, xi ≤ 1 (M p −mp ) (M q −mq )
(4.21)
4
for any x ∈ H with kxk = 1.
92
M.W. ALOMARI, S.S. DRAGOMIR
If m > 0, then by choosing g (t) = ln t we get the logarithmic inequality
1
M
q
q
|hA ln Ax, xi − hln Ax, xi hA x, xi| ≤ ln
(M q −mq )
(4.22)
4
m
for any x ∈ H with kxk = 1.
Remark 4.4. The interested reader in operator inequalities may be able to find
other recent results providing various bounds for the Čebyšev functional C (f, g; A; x)
in the papers [12]-[18] and the monograph [19].
References
1. M.W. Alomari and S.S. Dragomir, New Grüss type inequalities for the Stieltjes integral with
applications, submitted. Avalibale at [http://ajmaa.org/RGMIA/papers/v15/v15a.pdf].
2. G.A. Anastassiou, Grüss type inequalities for the Stieltjes integral, Nonlinear Funct. Anal.
Appl. 12 (2007), no. 4, 583–593.
3. G.A. Anastassiou, Chebyshev-Grüss type and comparison of integral means inequalities for
the Stieltjes integral, Panamer. Math. J. 17 (2007), no. 3, 91–109.
4. P. Cerone and S.S. Dragomir, New bounds for the Čebyšev functional, Appl. Math. Lett.
18 (2005) 603–611.
5. P. Cerone and S.S. Dragomir, A refinement of the Grüss inequality and applications,
Tamkang J. Math. 38 (2007), no. 1, 37–49.
6. S.S. Dragomir, Inequalities of Grüss type for the Stieltjes integral and applications, Kragujevac J. Math. 26 (2004) 89–112.
7. S.S. Dragomir, New Grüss’ type inequalities for functions of bounded variation and applications, Appl. Math. Lett. 25 (2012), no. 10, p. 1475–1479.
8. S.S. Dragomir, New estimates of the Čebyšev functional for Stieltjes integrals and applications, J. Korean Math. Soc. 41 (2004), no. 2, 249–264.
9. S.S. Dragomir, Sharp bounds of Čebyšev functional for Stieltjes integrals and applications,
Bull. Austral. Math. Soc. 67 (2003), no. 2, 257–266.
10. S.S. Dragomir, Some integral inequality of Grüss type, Indian J. Pure Appl. Math. 31
(2000), no. 4, 397–415.
11. S.S. Dragomir, Grüss type integral inequality for mappings of r-Hölder type and applications
for trapezoid formula, Tamkang J. Math. 31 (2000), no. 1, 43–47.
12. S.S. Dragomir, Čebyšev’s type inequalities for functions of selfadjoint operators in Hilbert
spaces, Linear Multilinear Algebra 58 (2010), no. 7, 805-814.
13. S.S. Dragomir, Grüss’ type inequalities for functions of selfadjoint operators in Hilbert
spaces, Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 11.
14. S.S. Dragomir, Some new Grüss’ type inequalities for functions of selfadjoint operators in
Hilbert spaces, Sarajevo J. Math. 18 (2010), 89–107.
15. S.S. Dragomir, Inequalities for the Čebyšev functional of two functions of selfadjoint operators in Hilbert spaces, Aust. J. Math. Anal. Appl. 6 (2009), No. 1, Art 7.
16. S.S. Dragomir, Some inequalities for the Čebyšev functional of two functions of selfadjoint
operators in Hilbert spaces, RGMIA Res. Rep. Coll. 11(e) (2008), Art. 8.
17. S.S. Dragomir, Quasi Grüss’ type inequalities for continuous functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. 13 (2010), Sup. Art. 12.
18. S.S. Dragomir, Grüss’ type inequalities for some classes of continuous functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. 13 (2010), no. 2, Art.
15.
19. S.S. Dragomir, Operator Inequalities of the Jensen, Čebyšev and Grüss Type. Springer Briefs
in Mathematics. Springer, New York, 2012.
20. S.S. Dragomir and I. Fedotov, An inequality of Grüss type for Riemann–Stieltjes integral
and applications for special means, Tamkang J. Math. 29 (1998), no. 4, 287–292.
NEW GRÜSS TYPE INEQUALITIES
93
Rb
1
21. G. Grüss, Über das maximum des absoluten Betrages von b−a
f (x) g (x) dx −
a
Rb
Rb
1
f (x) dx · a g (x) dx, Math. Z. 39 (1935), 215–226.
(b−a)2 a
22. G. Helmberg, Introduction to Spectral Theory in Hilbert Space, John Wiley & Sons, Inc.
-New York, 1969.
23. Z. Liu, Refinement of an inequality of Grüss type for Riemann–Stieltjes integral, Soochow
J. Math. 30 (2004), no. 4, 483–489.
1
Department of Mathematics, Faculty of Science, Jerash University, 26150
Jerash, Jordan.
E-mail address: mwomath@gmail.com
2
Mathematics, School of Engineering & Science, Victoria University, PO Box
14428, Melbourne City, MC 8001, Australia;
School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits, 2050, South Africa.
E-mail address: sever.dragomir@vu.edu.au