E extracta mathematicae Vol. 31, Núm. 1, 47 – 67 (2016)
Quasi Grüss Type Inequalities for Complex Functions
Defined on Unit Circle with Applications for Unitary
Operators in Hilbert Spaces
S. S. Dragomir
Mathematics, College of Engineering & Science, Victoria University,
PO Box 14428, Melbourne City, MC 8001, Australia
and
School of Computational & Applied Mathematics, University of the Witwatersrand,
Private Bag 3, Johannesburg 2050, South Africa
sever.dragomir@vu.edu.au http://rgmia.org/dragomir
Presented by Alfonso Montes Received April 14, 2015
Abstract: Some quasi Grüss type inequalities for the Riemann-Stieltjes integral of continuous
complex valued integrands defined on the complex unit circle C(0, 1) and various subclasses
of integrators are given. Natural applications for functions of unitary operators in Hilbert
spaces are provided.
Key words: Grüss type inequalities, Riemann-Stieltjes integral inequalities, Unitary opera-
tors in Hilbert spaces, Spectral theory, Quadrature rules.
AMS Subject Class. (2010): 26D15, 47A63.
1. Introduction
The concept of Riemann-Stieltjes integral
∫ b
a
f(t) du(t), where f is called
the integrand and u is called the integrator, plays an important role in Math-
ematics, for instance in the definition of complex integral, the representation
of bounded linear functionals on the Banach space of all continuous functions
on an interval [a, b], in the spectral representation of selfadjoint operators on
complex Hilbert spaces and other classes of operators such as the unitary
operators, etc.
One can approximate the Riemann-Stieltjes integral
∫ b
a
f(t) du(t) with the
following simpler quantity:
1
b − a
[u(b) − u(a)] ·
∫ b
a
f(t) dt ([11], [12]). (1.1)
In order to provide a priory sharp bounds for the approximation error,
47
48 s. s. dragomir
consider the functionals:
D(f, u; a, b) :=
∫ b
a
f(t) du(t) −
1
b − a
[u(b) − u(a)] ·
∫ b
a
f(t) dt.
If the integrand f is Riemann integrable on [a, b] and the integrator
u : [a, b] → R is L−Lipschitzian, i.e.,
|u(t) − u(s)| ≤ L|t − s| for each t, s ∈ [a, b], (1.2)
then the Riemann-Stieltjes integral
∫ b
a
f(t) du(t) exists and, as pointed out in
[11], the following quasi Grüss type inequality holds
|D(f, u; a, b)| ≤ L
∫ b
a
∣∣∣∣f(t) −
∫ b
a
1
b − a
f(s) ds
∣∣∣∣dt. (1.3)
The inequality (1.3) is sharp in the sense that the multiplicative constant
C = 1 in front of L cannot be replaced by a smaller quantity. Moreover,
if there exists the constants m, M ∈ R such that m ≤ f(t) ≤ M for a.e.
t ∈ [a, b], then [11]
|D(f, u; a, b)| ≤
1
2
L(M − m)(b − a). (1.4)
The constant 1
2
is best possible in (1.4).
We call this type of inequalities of quasi Grüss type since for integrators
of integral form u(t) := 1
b−a
∫ t
a
g(s)ds the left hand side becomes
∣∣∣∣ 1b − a
∫ b
a
f(t) du(t) −
1
b − a
∫ b
a
f(t) dt ·
1
b − a
∫ b
a
g(s) ds
∣∣∣∣
that is related with the well known Grüss inequality.
A different approach in the case of integrands of bounded variation were
considered by the same authors in 2001, [12], where they showed that
|D(f, u; a, b)| ≤ max
t∈[a,b]
∣∣∣∣f(t) − 1b − a
∫ b
a
f(s)ds
∣∣∣∣ b∨
a
(u), (1.5)
provided that f is continuous and u is of bounded variation. Here
∨b
a(u)
denotes the total variation of u on [a, b]. The inequality (1.5) is sharp.
quasi grüss type inequalities 49
If we assume that f is K−Lipschitzian, then [12]
|D(f, u; a, b)| ≤
1
2
K(b − a)
b∨
a
(u), (1.6)
with 1
2
the best possible constant in (1.6).
For various bounds on the error functional D(f, u; a, b) where f and u
belong to different classes of function for which the Stieltjes integral exists,
see [9], [8], [7], and [6] and the references therein.
For other inequalities for the Riemann-Stieltjes integral see [1] - [4], [5] -
[10], [14] and the references therein.
For continuous functions f : C(0, 1) → C, where C(0, 1) is the unit circle
from C centered in 0 and u : [a, b] ⊆ [0, 2π] → C is a function of bounded
variation on [a, b], we can define the following functional of quasi Grüss type
as well:
DC(f; u, a, b) :=
∫ b
a
f(eit)du(t) −
1
b − a
[u(b) − u(a)] ·
∫ b
a
f(eit)dt. (1.7)
In this paper we establish some bounds for the magnitude of SC(f; u, a, b)
when the integrand f : C(0, 1) → C satisfies some Hölder’s type conditions on
the circle C(0, 1) while the integrator u is of bonded variation.
It is shown that this functional can be naturally connected with continuous
functions of unitary operators on Hilbert spaces.
We recall here some basic facts on unitary operators and spectral families
that will be used in the sequel.
We say that the bounded linear operator U : H → H on the Hilbert space
H is unitary iff U∗ = U−1.
It is well known that (see for instance [13, p. 275-p. 276]), if U is a
unitary operator, then there exists a family of projections {Eλ}λ∈[0,2π], called
the spectral family of U with the following properties:
a) Eλ ≤ Eµ for 0 ≤ λ ≤ µ ≤ 2π;
b) E0 = 0 and E2π = 1H (the identity operator on H);
c) Eλ+0 = Eλ for 0 ≤ λ < 2π;
d) U =
∫ 2π
0
eiλdEλ, where the integral is of Riemann-Stieltjes type.
Moreover, if {Fλ}λ∈[0,2π] is a family of projections satisfying the require-
ments a) - d) above for the operator U, then Fλ = Eλ for all λ ∈ [0, 2π].
50 s. s. dragomir
Also, for every continuous complex valued function f : C(0, 1) → C on the
complex unit circle C(0, 1), we have
f(U) =
∫ 2π
0
f
(
eiλ
)
dEλ (1.8)
where the integral is taken in the Riemann-Stieltjes sense.
In particular, we have the equalities
⟨f(U)x, y⟩ =
∫ 2π
0
f
(
eiλ
)
d⟨Eλx, y⟩ (1.9)
and
∥f(U)x∥2 =
∫ 2π
0
∣∣f(eiλ)∣∣2d∥Eλx∥2 = ∫ 2π
0
∣∣f(eiλ)∣∣2d⟨Eλx, x⟩, (1.10)
for any x, y ∈ H.
From the above properties it follows that the function gx(λ) := ⟨Eλx, x⟩ is
monotonic nondecreasing and right continuous on [0, 2π] for any x ∈ H.
Such functions of unitary operators are
exp(U) =
∫ 2π
0
exp
(
eiλ
)
dEλ
and
Un =
∫ 2π
0
einλdEλ
for n an integer.
We can also define the trigonometric functions for a unitary operator U
by
sin(U) =
∫ 2π
0
sin
(
eiλ
)
dEλ and cos(U) =
∫ 2π
0
cos
(
eiλ
)
dEλ
and the hyperbolic functions by
sinh(U) =
∫ 2π
0
sinh
(
eiλ
)
dEλ and cosh(U) =
∫ 2π
0
cosh
(
eiλ
)
dEλ
where
sinh(z) :=
1
2
[exp z − exp(−z)] and cosh(z) :=
1
2
[exp z + exp(−z)], z ∈ C.
quasi grüss type inequalities 51
2. Inequalities for Riemann-Stieltjes integral
We say that the complex function f : C(0, 1) → C satisfies an H-r-Hölder’s
type condition on the circle C(0, 1), where H > 0 and r ∈ (0, 1] are given, if
|f(z) − f(w)| ≤ H|z − w|r (2.1)
for any w, z ∈ C(0, 1).
If r = 1 and L = H then we call it of L-Lipschitz type.
Consider the power function f : C\{0} → C, f(z) = zm where m is a
nonzero integer. Then, obviously, for any z, w belonging to the unit circle
C(0, 1) we have the inequality
|f(z) − f(w)| ≤ |m||z − w|
which shows that f is Lipschitzian with the constant L = |m| on the circle
C(0, 1).
For a ̸= ±1, 0 real numbers, consider the function f : C(0, 1) → C, fa(z) =
1
1−az . Observe that
|fa(z) − fa(w)| =
|a||z − w|
|1 − az||1 − aw|
(2.2)
for any z, w ∈ C(0, 1).
If z = eit with t ∈ [0, 2π], then we have
|1 − az|2 = 1 − 2a Re(z̄) + a2|z|2 = 1 − 2a cos t + a2
≥ 1 − 2|a| + a2 = (1 − |a|)2
therefore
1
|1 − az|
≤
1
|1 − |a||
and
1
|1 − aw|
≤
1
|1 − |a||
(2.3)
for any z, w ∈ C(0, 1).
Utilising (2.2) and (2.3) we deduce
|fa(z) − fa(w)| ≤
|a|
(1 − |a|)2
|z − w| (2.4)
for any z, w ∈ C(0, 1), showing that the function fa is Lipschitzian with the
constant La =
|a|
(1−|a|)2 on the circle C(0, 1).
52 s. s. dragomir
Theorem 1. Let f : C(0, 1) → C satisfies an H-r-Hölder’s type condition
on the circle C(0, 1), where H > 0 and r ∈ (0, 1] are given. If u : [a, b] ⊆
[0, 2π] → C is a function of bounded variation on [a, b], then
|DC(f; u, a, b)| ≤
2rH
b − a
max
t∈[a,b]
Br(a, b; t)
b∨
a
(u)
≤
H
r + 1
(b − a)r
b∨
a
(u)
(2.5)
where
Br(a, b; t) :=
∫ t
a
sinr
(
t − s
2
)
ds +
∫ b
t
sinr
(
s − t
2
)
ds
≤
1
2r
(t − a)r+1 + (b − t)r+1
r + 1
(2.6)
for any t ∈ [a, b].
In particular, if f is Lipschitzian with the constant L > 0, and [a, b] ⊂
[0, 2π] with b − a ̸= 2π, then we have the simpler inequality
|DC(f; u, a, b)| ≤
8L
b − a
sin2
(
b − a
4
) b∨
a
(u) ≤
1
2
L(b − a)
b∨
a
(u). (2.7)
If a = 0 and b = 2π and f is Lipschitzian with the constant L > 0, then
|DC(f; u, 0, 2π)| ≤
4L
π
2π∨
0
(u). (2.8)
Proof. We have
DC(f; u, a, b) =
∫ b
a
(
f
(
eit
)
−
1
b − a
∫ b
a
f
(
eis
)
ds
)
du(t)
=
1
b − a
∫ b
a
(∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
)
du(t).
(2.9)
It is known that if p : [c, d] → C is a continuous function and v : [c, d] → C is
of bounded variation, then the Riemann-Stieltjes integral
∫ d
c
p(t)dv(t) exists
and the following inequality holds∣∣∣∣
∫ d
c
p(t)dv(t)
∣∣∣∣ ≤ max
t∈[c,d]
|p(t)|
d∨
c
(v). (2.10)
quasi grüss type inequalities 53
Utilising this property and (2.9) we have
|DC(f; u, a, b)| =
1
b − a
∣∣∣∣
∫ b
a
(∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
)
du(t)
∣∣∣∣
≤
1
b − a
max
x∈[a,b]
∣∣∣∣
∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
∣∣∣∣ b∨
a
(u).
(2.11)
Utilising the properties of the Riemann integral and the fact that f is of
H-r-Hölder’s type on the circle C(0, 1) we have∣∣∣∣
∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
∣∣∣∣ ≤
∫ b
a
∣∣f(eit) − f(eis)∣∣ds
≤ H
∫ b
a
∣∣eis − eit∣∣r ds (2.12)
Since ∣∣eis − eit∣∣2 = ∣∣eis∣∣2 − 2 Re(ei(s−t)) + ∣∣eit∣∣2
= 2 − 2 cos(s − t) = 4 sin2
(
s − t
2
)
for any t, s ∈ R, then
∣∣eis − eit∣∣r = 2r ∣∣∣∣sin
(
s − t
2
)∣∣∣∣r (2.13)
for any t, s ∈ R.
Therefore∫ b
a
∣∣eis − eit∣∣r ds = 2r ∫ b
a
∣∣∣∣sin
(
s − t
2
)∣∣∣∣r ds
= 2r
[∫ t
a
sinr
(
t − s
2
)
ds +
∫ b
t
sinr
(
s − t
2
)
ds
] (2.14)
for any t ∈ [a, b].
On making use of (2.12) and (2.14) we have
max
x∈[a,b]
∣∣∣∣
∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
∣∣∣∣ ≤ 2rH max
t∈[a,b]
Br(a, b; t)
and the first inequality in (2.5) is proved.
54 s. s. dragomir
Utilising the elementary inequality | sin(x)| ≤ |x|, x ∈ R we have
Br(a, b; t) ≤
∫ t
a
(
t − s
2
)r
ds +
∫ b
t
(
s − t
2
)r
ds
=
1
2r
(t − a)r+1 + (b − t)r+1
r + 1
(2.15)
for any t ∈ [a, b], and the inequality (2.6) is proved.
If we consider the auxiliary function φ : [a, b] → R,
φ(t) = (t − a)r+1 + (b − t)r+1, r ∈ (0, 1]
then
φ′(t) = (r + 1) [(t − a)r − (b − t)r]
and
φ′′(t) = (r + 1)r[(t − a)r−1 + (b − t)r−1].
We have φ′(t) = 0 iff t = a+b
2
, φ′(t) < 0 for t ∈
(
a, a+b
2
)
and φ′(t) > 0 for
t ∈ (a+b
2
, b). We also have that φ′′(t) > 0 for any t ∈ (a, b) showing that φ
is strictly decreasing on
(
a, a+b
2
)
and strictly increasing on (a+b
2
, b). We also
have that
min
t∈[a,b]
φ(t) = φ
(
a + b
2
)
=
(b − a)r+1
2r
and
max
t∈[a,b]
φ(t) = φ(a) = φ(b) = (b − a)r+1.
Taking the maximum over t ∈ [a, b] in (2.15) we deduce the second inequality
in (2.5).
For r = 1 we have
B(a, b; t) :=
∫ t
a
sin
(
t − s
2
)
ds +
∫ b
t
sin
(
s − t
2
)
ds
= 2 − 2 cos
(
t − a
2
)
− 2 cos
(
b − t
2
)
+ 2
= 2
[
1 − cos
(
t − a
2
)
+ 1 − cos
(
b − t
2
)]
= 2
[
2 sin2
(
t − a
4
)
+ 2 sin2
(
b − t
4
)]
= 4
[
sin2
(
t − a
4
)
+ sin2
(
b − t
4
)]
quasi grüss type inequalities 55
for any t ∈ [a, b].
Now, if we take the derivative in the first equality, we have
B′(a, b; t) = sin
(
t − a
2
)
− sin
(
b − t
2
)
= 2 sin
(
t − a+b
2
2
)
cos
(
b − a
4
)
,
for [a, b] ⊂ [0, 2π] and b − a ̸= 2π.
We observe that B′(a, b; t) = 0 iff t = a+b
2
, B′(a, b; t) < 0 for t ∈
(
a, a+b
2
)
and B′(a, b; t) > 0 for t ∈ (a+b
2
, b). The second derivative is given by
B′′(a, b; t) = cos
(
t − a+b
2
2
)
cos
(
b − a
4
)
and we observe that B′′(a, b; t) > 0 for t ∈ (a, b).
Therefore the function B(a, b; ·) is strictly decreasing on
(
a, a+b
2
)
and
strictly increasing on
(
a+b
2
, b
)
. It is also a strictly convex function on (a, b).
We have
min
t∈[a,b]
B(a, b; t) = B
(
a, b;
a + b
2
)
= 8 sin2
(
b − a
8
)
and
max
t∈[a,b]
B(a, b; t) = B(a, b; a) = B(a, b; b) = 4 sin2
(
b − a
4
)
.
This proves the bound (2.7).
If a = 0 and b = 2π, then
B(0, 2π; t) = 4
[
sin2
(
t
4
)
+ sin2
(
2π − t
4
)]
= 4
and by (2.5) we get (2.8).
The proof is complete.
The following result also holds:
Theorem 2. Let f : C(0, 1) → C satisfies an H-r-Hölder’s type condition
on the circle C(0, 1), where H > 0 and r ∈ (0, 1] are given. If u : [a, b] ⊆
[0, 2π] → C is a function of Lipschitz type with the constant K > 0 on [a, b],
then
|DC(f; u, a, b)| ≤
2rHK
b − a
Cr(a, b) ≤
2HK(b − a)r+1
(r + 1)(r + 2)
(2.16)
56 s. s. dragomir
where
Cr(a, b) :=
∫ b
a
∫ t
a
sinr
(
t − s
2
)
ds dt +
∫ b
a
∫ b
t
sinr
(
s − t
2
)
ds dt
≤
(b − a)r+2
2r−1(r + 1)(r + 2)
.
(2.17)
In particular, if f is Lipschitzian with the constant L > 0, then we have
the simpler inequality
|DC(f; u, a, b)| ≤
16LK
b − a
[
b − a
2
− sin
(
b − a
2
)]
≤
LK(b − a)2
3
.
(2.18)
Proof. It is well known that if p : [c, d] → C is a Riemann integrable
function and v : [c, d] → C is Lipschitzian with the constant M > 0, then
the Riemann-Stieltjes integral
∫ d
c
p(t)dv(t) exists and the following inequality
holds ∣∣∣∣
∫ d
c
p(t)dv(t)
∣∣∣∣ ≤ M
∫ d
c
|p(t)|dt. (2.19)
Utilising the equality (2.9) and this property we have
|DC(f; u, a, b)| =
1
b − a
∣∣∣∣
∫ b
a
(∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
)
du(t)
∣∣∣∣
≤
K
b − a
∫ b
a
∣∣∣∣
(∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
)∣∣∣∣dt.
(2.20)
From (2.12) and (2.14) we have∣∣∣∣
∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
∣∣∣∣
≤
∫ b
a
∣∣f (eit) − f (eis)∣∣ds
≤ H
∫ b
a
∣∣eis − eit∣∣r ds
= 2rH
[∫ t
a
sinr
(
t − s
2
)
ds +
∫ b
t
sinr
(
s − t
2
)
ds
]
(2.21)
quasi grüss type inequalities 57
and by (2.20) we deduce the first part of (2.16).
Since, by (2.15), we have
∫ t
a
(
t − s
2
)r
ds +
∫ b
t
(
s − t
2
)r
ds =
1
2r
(t − a)r+1 + (b − t)r+1
r + 1
,
then
Cr(a, b) ≤
∫ b
a
[∫ t
a
(
t − s
2
)r
ds +
∫ b
t
(
s − t
2
)r
ds
]
dt
≤
1
2r
∫ b
a
(t − a)r+1 + (b − t)r+1
r + 1
dt
=
(b − a)r+2
2r−1(r + 1)(r + 2)
,
which proves the inequality (2.17).
For r = 1, we have
C1(a, b) :=
∫ b
a
[∫ t
a
sin
(
t − s
2
)
ds +
∫ b
t
sin
(
s − t
2
)
ds
]
dt
=
∫ b
a
[
2 − 2 cos
(
t − a
2
)
− 2 cos
(
b − t
2
)
+ 2
]
dt
= 4(b − a) − 4 sin
(
b − a
2
)
− 4 sin
(
b − a
2
)
= 8
[
b − a
2
− sin
(
b − a
2
)]
,
which, by (2.16), produces the desired inequality (2.18).
Remark 1. In the case b = 2π and a = 0 the inequality (2.18) produces
the simple inequality
|DC(f; u, 0, 2π)| ≤ 8LK. (2.22)
58 s. s. dragomir
The following result for monotonic integrators also holds.
Theorem 3. Let f : C(0, 1) → C satisfies an H-r-Hölder’s type condition
on the circle C(0, 1), where H > 0 and r ∈ (0, 1] are given. If u : [a, b] ⊆
[0, 2π] → R is a monotonic nondecreasing function on [a, b], then
|DC(f; u, a, b)| ≤
2rH
b − a
Dr(a, b)
≤
H
(r + 1)(b − a)
∫ b
a
[
(t − a)r+1 + (b − t)r+1
]
du(t)
≤
H
(r + 1)
(b − a)r[u(b) − u(a)]
(2.23)
where
Dr(a, b) :=
∫ b
a
Br(a, b; t)du(t) (2.24)
and Br(a, b; t) is given by (2.6).
In particular, if f is Lipschitzian with the constant L > 0, then we have
the simpler inequality
|DC(f; u, a, b)| ≤
8L
b − a
∫ b
a
[
sin2
(
t − a
4
)
+ sin2
(
b − t
4
)]
du(t)
≤
L
2
(b − a)[u(b) − u(a)].
(2.25)
Proof. It is well known that if p : [c, d] → C is a continuous function and
v : [c, d] → R is monotonic nondecreasing on [c, d], then the Riemann-Stieltjes
integral
∫ d
c
p(t)dv(t) exists and the following inequality holds
∣∣∣∣
∫ d
c
p(t)dv(t)
∣∣∣∣ ≤
∫ d
c
|p(t)|dv(t). (2.26)
Utilising this property and the identity (2.9) we have
quasi grüss type inequalities 59
|DC(f; u, a, b)|
=
1
b − a
∣∣∣∣
∫ b
a
(∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
)
du(t)
∣∣∣∣
≤
1
b − a
∫ b
a
∣∣∣∣
(∫ b
a
[
f
(
eit
)
− f
(
eis
)]
ds
)∣∣∣∣du(t)
≤
1
b − a
∫ b
a
(∫ b
a
∣∣(f (eit) − f (eis))∣∣ds)du(t)
≤
H
b − a
∫ b
a
(∫ b
a
∣∣eis − eit∣∣r ds)du(t)
=
2rH
b − a
∫ b
a
[∫ t
a
sinr
(
t − s
2
)
ds +
∫ b
t
sinr
(
s − t
2
)
ds
]
du(t).
(2.27)
We also have that∫ b
a
[∫ t
a
sinr
(
t − s
2
)
ds +
∫ b
t
sinr
(
s − t
2
)
ds
]
du(t)
≤
∫ b
a
[∫ t
a
(
t − s
2
)r
ds +
∫ b
t
(
s − t
2
)r
ds
]
du(t)
=
1
2r
∫ b
a
(t − a)r+1 + (b − t)r+1
r + 1
du(t)
=
1
2r(r + 1)
∫ b
a
[(t − a)r+1 + (b − t)r+1]du(t)
and the first part of the inequality (2.23) is proved.
Since
max
t∈[a,b]
[
(t − a)r+1 + (b − t)r+1
]
= (b − a)r+1
then the last part of (2.23) is also proved
For r = 1 we have
D1(a, b) :=
∫ b
a
B1(a, b; t)du(t)
= 4
∫ b
a
[
sin2
(
t − a
4
)
+ sin2
(
b − t
4
)]
du(t)
and the inequality (2.25) is obtained.
60 s. s. dragomir
Remark 2. The case a = 0, b = 2π can be stated as
|DC(f; u, 0, 2π)| ≤
4L
π
[u(2π) − u(0)]. (2.28)
Indeed, by (2.25) we have
|DC(f; u, 0, 2π)| ≤
8L
2π
∫ 2π
0
[
sin2
(
t
4
)
+ sin2
(
2π − t
4
)]
du(t)
=
4L
π
∫ 2π
0
[
sin2
(
t
4
)
+ sin2
(
π
2
−
t
4
)]
du(t)
=
4L
π
∫ 2π
0
[
sin2
(
t
4
)
+ cos2
(
t
4
)]
du(t)
=
4L
π
[u(2π) − u(0)].
3. Applications for functions of unitary operators
We have the following vector inequality for functions of unitary operators.
Theorem 4. Assume that f : C(0, 1) → C satisfies an L-Lipschitz type
condition on the circle C(0, 1), where L > 0 is given. If the operator
U : H → H on the Hilbert space H is unitary and {Eλ}λ∈[0,2π] is its spectral
family, then∣∣∣∣⟨f(U)x, y⟩ − 12π
∫ 2π
0
f(eit)dt · ⟨x, y⟩
∣∣∣∣
≤
4L
π
2π∨
0
(⟨E(·)x, y⟩) ≤
4L
π
∥x∥∥y∥
(3.1)
for any x, y ∈ H.
Proof. For given x, y ∈ H, define the function u(λ) := ⟨Eλx, y⟩, λ ∈ [0, 2π].
We will show that u is of bounded variation and
2π∨
0
(u) =:
2π∨
0
(
⟨E(·)x, y⟩
)
≤ ∥x∥∥y∥. (3.2)
It is well known that, if P is a nonnegative selfadjoint operator on H, i.e.,
⟨Px, x⟩ ≥ 0 for any x ∈ H, then the following inequality is a generalization of
quasi grüss type inequalities 61
the Schwarz inequality in H
|⟨Px, y⟩|2 ≤ ⟨Px, x⟩⟨Py, y⟩, (3.3)
for any x, y ∈ H.
Now, if d : 0 = t0 < t1 < · · · < tn−1 < tn = 2π is an arbitrary partition
of the interval [0, 2π], then we have by Schwarz’s inequality for nonnegative
operators (3.3) that
2π∨
0
(⟨
E(·)x, y
⟩)
= sup
d
{
n−1∑
i=0
∣∣⟨(Eti+1 − Eti)x, y⟩∣∣
}
≤ sup
d
{
n−1∑
i=0
[⟨(
Eti+1 − Eti
)
x, x
⟩1/2 ⟨(
Eti+1 − Eti
)
y, y
⟩1/2]}
:= I.
(3.4)
By the Cauchy-Buniakovski-Schwarz inequality for sequences of real numbers
we also have that
I ≤ sup
d
[
n−1∑
i=0
⟨(
Eti+1 − Eti
)
x, x
⟩]1/2 [n−1∑
i=0
⟨(
Eti+1 − Eti
)
y, y
⟩]1/2
≤ sup
d
[
n−1∑
i=0
⟨(
Eti+1 − Eti
)
x, x
⟩]1/2
sup
d
[
n−1∑
i=0
⟨(
Eti+1 − Eti
)
y, y
⟩]1/2
=
[
2π∨
0
(⟨
E(·)x, x
⟩)]1/2 [2π∨
0
(⟨
E(·)y, y
⟩)]1/2
= ∥x∥∥y∥
(3.5)
for any x, y ∈ H.
Utilising the inequality (2.8) we can write that∣∣∣∣
∫ 2π
0
f(eit) d⟨Etx, y⟩ −
1
2π
[⟨E2πx, y⟩ − ⟨E0x, y⟩] ·
∫ 2π
0
f(eit)dt
∣∣∣∣
≤
4L
π
2π∨
0
(⟨
E(·)x, y
⟩)
,
(3.6)
for any x, y ∈ H.
On making use of the representation theorem (1.9) and the inequality (3.2)
we deduce the desired result (3.1).
62 s. s. dragomir
Remark 3. Consider the function f : C(0, 1) → C, fa(z) = 11−az with a
real and 0 < |a| < 1. We know that this function is Lipschitzian with the
constant L =
|a|
(1−|a|)2 . Since |ae
it| = |a| < 1, then∫ 2π
0
f
(
eit
)
dt =
∫ 2π
0
1
1 − aeit
dt =
∫ 2π
0
∞∑
n=0
(
aeit
)n
dt
=
∞∑
n=0
an
∫ 2π
0
(
eit
)n
dt =
∫ 2π
0
dt = 2π,
since for any natural number n ≥ 1 we have
∫ 2π
0
(eit)ndt = 0.
Applying the inequality (3.1) we have∣∣∣⟨(1H − aU)−1 x, y⟩ − ⟨x, y⟩∣∣∣
≤
4|a|
π(1 − |a|)2
2π∨
0
(⟨
E(·)x, y
⟩)
≤
4|a|
π(1 − |a|)2
∥x∥∥y∥
(3.7)
for any x, y ∈ H.
4. A quadrature rule
We consider the following partition of the interval [a, b]
∆n : a = x0 < x1 < · · · < xn−1 < xn = b.
Define hk := xk+1 − xk, 0 ≤ k ≤ n − 1 and ν(∆n) = max{hk : 0 ≤ k ≤ n − 1}
the norm of the partition ∆n.
For the continuous function f : C(0, 1) → C and the function u : [a, b] ⊆
[0, 2π] → C of bounded variation on [a, b], define the quadrature rule
Dn(f, u, ∆n) :=
n−1∑
k=0
u(xk+1) − u(xk)
xk+1 − xk
∫ xk+1
xk
f
(
eit
)
dt (4.1)
and the remainder Rn(f, u, ∆n) in approximating the Riemann-Stieltjes inte-
gral
∫ b
a
f(eit)du(t) by Dn(f, u, ∆n). Then we have∫ b
a
f
(
eit
)
du(t) = Dn(f, u, ∆n) + Rn(f, u, ∆n). (4.2)
The following result provides a priory bounds for Rn(f, u, ∆n) in several in-
stances of f and u as above.
quasi grüss type inequalities 63
Proposition 1. Assume that f : C(0, 1) → C satisfies the following Lip-
schitz type condition
|f(z) − f(w)| ≤ L|z − w|
for any w, z ∈ C(0, 1), where L > 0 is given given.
If [a, b] ⊆ [0, 2π] and the function u : [a, b] → C is of bounded variation on
[a, b], then for any partition ∆n : a = x0 < x1 < · · · < xn−1 < xn = b with
the norm ν(∆n) < 2π we have the error bound
|Rn(f, u, ∆n)| ≤ 8L
n−1∑
k=0
1
xk+1 − xk
sin2
(
xk+1 − xk
4
)xk+1∨
xk
(u)
≤
1
2
Lν(∆n)
b∨
a
(u).
(4.3)
Proof. Since ν(∆n) < 2π, then on writing inequality (2.7) on each interval
[xk, xk+1], where 0 ≤ k ≤ n − 1, we have∣∣∣∣
∫ xk+1
xk
f
(
eit
)
du(t) −
u(xk+1) − u(xk)
xk+1 − xk
∫ xk+1
xk
f
(
eit
)
dt
∣∣∣∣
≤
8L
xk+1 − xk
sin2
(
xk+1 − xk
4
)xk+1∨
xk
(u).
Utilising the generalized triangle inequality we then have
|Rn(f, u, ∆n)|
=
∣∣∣∣∣
n−1∑
k=0
[∫ xk+1
xk
f
(
eit
)
du(t) −
u(xk+1) − u(xk)
xk+1 − xk
∫ xk+1
xk
f
(
eit
)
dt
]∣∣∣∣∣
≤
n−1∑
k=0
∣∣∣∣
[∫ xk+1
xk
f
(
eit
)
du(t) −
u(xk+1) − u(xk)
xk+1 − xk
∫ xk+1
xk
f
(
eit
)
dt
]∣∣∣∣
≤
n−1∑
k=0
8L
xk+1 − xk
sin2
(
xk+1 − xk
4
)xk+1∨
xk
(u)
≤ 8L max
0≤k≤n−1
{
1
xk+1 − xk
sin2
(
xk+1 − xk
4
)}n−1∑
k=0
xk+1∨
xk
(u)
= 8L max
0≤k≤n−1
{
1
xk+1 − xk
sin2
(
xk+1 − xk
4
)} b∨
a
(u).
64 s. s. dragomir
Since
1
xk+1 − xk
sin2
(
xk+1 − xk
4
)
≤
1
16
(xk+1 − xk)
then
max
0≤k≤n−1
{
1
xk+1 − xk
sin2
(
xk+1 − xk
4
)}
≤
1
16
ν(∆n)
and the last part of (4.3) also holds.
Remark 4. The above proposition has some particular cases of interest. If
we take for instance a = 0, x1 = π and b = 2π, then we have from (4.3) that∣∣∣∣
∫ 2π
0
f
(
eit
)
du(t) −
u(π) − u(0)
π
∫ π
0
f
(
eit
)
dt −
u(2π) − u(π)
π
∫ 2π
π
f(eit)dt
∣∣∣∣
≤
8L
π
2π∨
0
(u).
Remark 5. We observe that the last bound in (4.3) provides a simple way
to choose a division such that the accuracy in approximation is better that a
given small ε > 0. Indeed, if we want
1
2
Lν(∆n)
b∨
a
(u) ≤ ε
then we need to take ∆n such that
ν(∆n) ≤
2ε∨b
a(u)L
.
The above proposition can be also utilized to approximate functions of
unitary operators as follows.
We consider the following partition of the interval [0, 2π]
Γn : 0 = λ0 < λ1 < · · · < λn−1 < λn = 2π
where 0 ≤ k ≤ n − 1.
If U is a unitary operator on the Hilbert space H and {Eλ}λ∈[0,2π], the
spectral family of U, then we can introduce the following sums:
Dn(f, U,Γn; x, y)
:=
n−1∑
k=0
1
λk+1 − λk
∫ λk+1
λk
f
(
eit
)
dt ·
⟨(
Eλk+1 − Eλk
)
x, y
⟩
.
(4.4)
quasi grüss type inequalities 65
Corollary 1. Assume that f : C(0, 1) → C satisfies the following Lips-
chitz type condition
|f(z) − f(w)| ≤ L|z − w|
for any w, z ∈ C(0, 1), where L > 0 is given. Assume also that U is a unitary
operator on the Hilbert space H and {Eλ}λ∈[0,2π] is the spectral family of U.
If Γn is a partition of the interval [0, 2π] with ν(Γn) < 2π then we have
the representation
⟨f(U)x, y⟩ = Dn(f, U, Γn; x, y) + Rn(f, U, Γn; x, y) (4.5)
with the error Rn(f, U, ∆n; x, y) satisfying the bounds
|Rn(f, U, Γn; x, y)|
≤ 8L
n−1∑
k=0
1
λk+1 − λk
sin2
(
λk+1 − λk
4
)λk+1∨
λk
(⟨
E(·)x, y
⟩)
≤
1
2
Lν(Γn)
2π∨
0
(⟨
E(·)x, y
⟩)
≤
1
2
Lν(Γn)∥x∥∥y∥
(4.6)
for any x, y ∈ H.
Remark 6. Consider the exponential mean
Ez(p, q) :=
exp(pz) − exp(qz)
p − q
defined for complex numbers z and the real numbers p, q with p ̸= q.
For the function f(z) = zm with m an integer we have∫ p
q
f
(
eit
)
dt =
∫ p
q
eimtdt =
1
im
(
eimp − eimq
)
=
1
im
(p − q)Eeim(p, q).
For a partition Γn as above, define the sum
Pn(U, Γn; x, y) :=
1
im
n−1∑
k=0
Eeim(λk+1, λk)
⟨(
Eλk+1 − Eλk
)
x, y
⟩
. (4.7)
We can approximate the power m of an unitary operator as follows:
⟨Umx, y⟩ = Pn(U, Γn; x, y) + Tn(U, Γn; x, y) (4.8)
66 s. s. dragomir
where the error Tn(U, Γn; x, y) satisfies the bounds
|Tn(U, Γn; x, y)|
≤ 8|m|
n−1∑
k=0
1
λk+1 − λk
sin2
(
λk+1 − λk
4
)λk+1∨
λk
(⟨
E(·)x, y
⟩)
≤
1
2
|m|ν(Γn)
2π∨
0
(⟨
E(·)x, y
⟩)
≤
1
2
|m|ν(Γn)∥x∥∥y∥
(4.9)
for any vectors x, y ∈ H.
References
[1] M.W. Alomari, A companion of Ostrowski’s inequality for the Riemann–
Stieltjes integral
∫ b
a
f(t)du(t), where f is of bounded variation and u is of r-
H-Hölder type and applications, Appl. Math. Comput. 219 (9) (2013), 4792 –
4799.
[2] M.W. Alomari, Some Grüss type inequalities for Riemann-Stieltjes integral
and applications, Acta Math. Univ. Comenian. (N.S.) 81 (2) (2012), 211 –
220.
[3] G.A. Anastassiou, Grüss type inequalities for the Stieltjes integral, Nonlin-
ear Funct. Anal. Appl. 12 (4) (2007), 583 – 593.
[4] G.A. Anastassiou, A new expansion formula, Cubo Mat. Educ. 5 (1) (2003),
25 – 31.
[5] N.S. Barnett, W.S. Cheung, S.S. Dragomir, A. Sofo, Ostrowski
and trapezoid type inequalities for the Stieltjes integral with Lipschitzian
integrands or integrators, Comput. Math. Appl. 57 (2) (2009), 195 – 201.
[6] W.S. Cheung, S.S. Dragomir, Two Ostrowski type inequalities for the
Stieltjes integral of monotonic functions, Bull. Austral. Math. Soc., 75 (2)
(2007), 299 – 311.
[7] S.S. Dragomir, Inequalities of Grüss type for the Stieltjes integral, Kragu-
jevac J. Math., 26 (2004), 89 – 122.
[8] S.S. Dragomir, A generalisation of Cerone’s identity and applications, Tam-
sui Oxf J. Math., 23 (1) (2007), 79 – 90.
[9] S.S. Dragomir, Inequalities for Stieltjes integrals with convex integrators
and applications, Appl. Math. Lett., 20 (2) (2007), 123 – 130.
[10] S.S. Dragomir, C. BuŞE, M.V. Boldea, L. Braescu, A generaliza-
tion of the trapezoidal rule for the Riemann-Stieltjes integral and applica-
tions, Nonlinear Anal. Forum, (Korea) 6 (2) (2001), 337 – 351.
[11] S.S. Dragomir, I.A. Fedotov, An inequality of Grüss type for the
Riemann-Stieltjes integral and applications for special means, Tamkang J.
Math., 29 (4) (1998), 287 – 292.
quasi grüss type inequalities 67
[12] S.S. Dragomir, I. Fedotov, A Grüss type inequality for mappings of
bounded variation and applications to numerical analysis, Nonlinear Funct.
Anal. Appl., 6 (3) (2001), 425 – 433.
[13] G. Helmberg, “Introduction to Spectral Theory in Hilbert Space”, John
Wiley, New York, 1969.
[14] Z. Liu, Refinement of an inequality of Grüss type for Riemann-Stieltjes integral,
Soochow J. Math., 30 (4) (2004), 483 – 489.