Journal of Inequalities in Pure and
Applied Mathematics
REVERSES OF THE CONTINUOUS TRIANGLE INEQUALITY
FOR BOCHNER INTEGRAL OF VECTOR-VALUED FUNCTIONS
IN HILBERT SPACES
volume 6, issue 2, article 46,
2005.
S.S. DRAGOMIR
School of Computer Science and Mathematics
Victoria University of Technology
PO Box 14428, MCMC 8001
VIC, Australia.
EMail: sever@csm.vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Received 07 June, 2004;
accepted 15 April, 2005.
Communicated by: R.N. Mohapatra
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
115-04
Quit
Abstract
Some reverses of the continuous triangle inequality for Bochner integral of
vector-valued functions in Hilbert spaces are given. Applications for complexvalued functions are provided as well.
2000 Mathematics Subject Classification: 46C05, 26D15, 26D10.
Key words: Triangle inequality, Reverse inequality, Hilbert spaces, Bochner integral.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Reverses for a Unit Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Reverses for Orthonormal Families of Vectors . . . . . . . . . . . . 12
4
Applications for Complex-Valued Functions . . . . . . . . . . . . . . 18
References
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
1.
Introduction
Let f : [a, b] → K, K = C or R be a Lebesgue integrable function. The
following inequality, which is the continuous version of the triangle inequality,
Z b
Z b
(1.1)
f (x) dx ≤
|f (x)| dx,
a
a
plays a fundamental role in Mathematical Analysis and its applications.
It appears, see [4, p. 492], that the first reverse inequality for (1.1) was
obtained by J. Karamata in his book from 1949, [2]. It can be stated as
Z b
Z b
(1.2)
cos θ
|f (x)| dx ≤ f (x) dx ,
a
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
a
Title Page
provided
−θ ≤ arg f (x) ≤ θ, x ∈ [a, b]
π
Contents
for given θ ∈ 0, 2 .
This integral inequality is the continuous version of a reverse inequality for
the generalised triangle inequality
n
n
X
X
(1.3)
cos θ
|zi | ≤ zi ,
JJ
J
provided
Page 3 of 23
i=1
i=1
a − θ ≤ arg (zi ) ≤ a + θ, for i ∈ {1, . . . , n} ,
II
I
Go Back
Close
Quit
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
where a ∈ R and θ ∈ 0, π2 , which, as pointed out in [4, p. 492], was first
discovered by M. Petrovich in 1917, [5], and, subsequently rediscovered by
other authors, including J. Karamata [2, p. 300 – 301], H.S. Wilf [6], and in an
equivalent form, by M. Marden [3].
The first to consider the problem in the more general case of Hilbert and
Banach spaces, were J.B. Diaz and F.T. Metcalf [1] who showed that, in an
inner product space H over the real or complex number field, the following
reverse of the triangle inequality holds
n
n
X X
(1.4)
r
kxi k ≤ xi ,
i=1
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
i=1
S.S. Dragomir
provided
0≤r≤
Re hxi , ai
,
kxi k
i ∈ {1, . . . , n} ,
and a ∈ H is a unit vector, i.e., kak = 1. The case of equality holds in (1.4) if
and only if
!
n
n
X
X
(1.5)
xi = r
kxi k a.
i=1
i=1
The main aim of this paper is to point out some reverses of the triangle
inequality for Bochner integrable functions f with values in Hilbert spaces and
defined on a compact interval [a, b] ⊂ R. Applications for Lebesgue integrable
complex-valued functions are provided as well.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
2.
Reverses for a Unit Vector
We recall that f ∈ L ([a, b] ; H) , the space of Bochner integrable functions with
values in a Hilbert space H, if and only if f : [a, b] → H is Bochner measurable
Rb
on [a, b] and the Lebesgue integral a kf (t)k dt is finite.
The following result holds:
Theorem 2.1. If f ∈ L ([a, b] ; H) is such that there exists a constant K ≥ 1
and a vector e ∈ H, kek = 1 with
(2.1)
kf (t)k ≤ K Re hf (t) , ei for a.e. t ∈ [a, b] ,
then we have the inequality:
Z b
Z b
(2.2)
kf (t)k dt ≤ K f (t) dt
.
a
a
The case of equality holds in (2.2) if and only if
Z b
Z b
1
(2.3)
f (t) dt =
kf (t)k dt e.
K
a
a
Proof. By the Schwarz inequality in inner product spaces, we have
Z b
Z b
f (t) dt = (2.4)
f (t) dt
kek
a
a
Z b
≥ f (t) dt, e a
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
Z b
≥ Re
f (t) dt, e a
Z b
Z b
≥ Re
f (t) dt, e =
Re hf (t) , ei dt.
a
a
From the condition (2.1), on integrating over [a, b] , we deduce
Z b
Z
1 b
(2.5)
Re hf (t) , ei dt ≥
kf (t)k dt,
K a
a
and thus, on making use of (2.4) and (2.5), we obtain the desired inequality
(2.2).
If (2.3) holds true, then, obviously
Z b
Z b
Z b
K
f (t) dt = kek
kf (t)k dt =
kf (t)k dt,
a
a
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
a
showing that (2.2) holds with equality.
If we assume that the equality holds in (2.2), then by the argument provided
at the beginning of our proof, we must have equality in each of the inequalities
from (2.4) and (2.5).
Observe that in Schwarz’s inequality kxk kyk ≥ Re hx, yi , x, y ∈ H, the
case of equality holds if and only if there exists a positive scalarR µ such that x =
b
µe. Therefore, equality holds in the first inequality in (2.4) iff a f (t) dt = λe,
with λ ≥ 0 .
If we assume that a strict
R b inequality holds
R b in (2.1) on a subset of nonzero
Lebesgue measures, then a kf (t)k dt < K a Re hf (t) , ei dt, and by (2.4) we
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
deduce a strict inequality in (2.2), which contradicts the assumption. Thus, we
must have kf (t)k = K Re hf (t) , ei for a.e. t ∈ [a, b] .
If we integrate this equality, we deduce
Z b
Z b
kf (t)k dt = K
Re hf (t) , ei dt
a
a
Z b
= K Re
f (t) dt, e
a
= K Re hλe, ei = λK,
giving
Z
1 b
λ=
kf (t)k dt,
K a
and thus the equality (2.3) is necessary.
This completes the proof.
A more appropriate result from an applications point of view is perhaps the
following result.
Corollary 2.2. Let e be a unit vector in the Hilbert space (H; h·, ·i) , ρ ∈ (0, 1)
and f ∈ L ([a, b] ; H) so that
(2.6)
kf (t) − ek ≤ ρ for a.e. t ∈ [a, b] .
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Then we have the inequality
(2.7)
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
Z b
Z b
p
,
1 − ρ2
kf (t)k dt ≤ f
(t)
dt
a
a
Page 7 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
with equality if and only if
Z b
Z b
p
2
(2.8)
f (t) dt = 1 − ρ
kf (t)k dt · e.
a
a
Proof. From (2.6), we have
kf (t)k2 − 2 Re hf (t) , ei + 1 ≤ ρ2 ,
giving
kf (t)k2 + 1 − ρ2 ≤ 2 Re hf (t) , ei
for a.e. t ∈ [a, b]p
.
Dividing by 1 − ρ2 > 0, we deduce
(2.9)
S.S. Dragomir
p
kf (t)k2
2 Re hf (t) , ei
p
+ 1 − ρ2 ≤ p
1 − ρ2
1 − ρ2
for a.e. t ∈ [a, b] .
On the other hand, by the elementary inequality
p
√
+ qα ≥ 2 pq, p, q ≥ 0, α > 0
α
we have
Title Page
Contents
JJ
J
II
I
Go Back
Close
2
(2.10)
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
p
kf (t)k
2 kf (t)k ≤ p
+ 1 − ρ2
1 − ρ2
Quit
Page 8 of 23
for each t ∈ [a, b] .
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
Making use of (2.9) and (2.10), we deduce
1
kf (t)k ≤ p
1 − ρ2
for a.e. t ∈ [a, b] .
Applying Theorem 2.1 for K = √ 1
Re hf (t) , ei
1−ρ2
, we deduce the desired inequality
(2.7).
In the same spirit, we also have the following corollary.
Corollary 2.3. Let e be a unit vector in H and M ≥ m > 0. If f ∈ L ([a, b] ; H)
is such that
(2.11)
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Re hM e − f (t) , f (t) − mei ≥ 0 for a.e. t ∈ [a, b] ,
Title Page
or, equivalently,
M +m 1
(2.12)
e
≤ (M − m) for a.e. t ∈ [a, b] ,
f (t) −
2
2
then we have the inequality
√
Z b
Z
2 mM b
,
(2.13)
kf (t)k dt ≤ f
(t)
dt
M +m a
a
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
or, equivalently,
Z b
(0 ≤)
kf (t)k dt − f (t) dt
a
a
√
√ 2 Z
M− m b
f (t) dt
≤
.
M +m
Z
(2.14)
b
a
The equality holds in (2.13) (or in the second part of (2.14)) if and only if
√
Z b
Z b
2 mM
kf (t)k dt e.
(2.15)
f (t) dt =
M +m
a
a
Proof. Firstly, we remark that if x, z, Z ∈ H, then the following statements are
equivalent
(i) Re hZ − x, x − zi ≥ 0
and
1
≤ kZ − zk .
(ii) x − Z+z
2
2
Using this fact, we may simply realise that (2.9) and (2.10) are equivalent.
Now, from (2.9), we obtain
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
kf (t)k2 + mM ≤ (M + m) Re hf (t) , ei
√
for a.e. t ∈ [a, b] . Dividing this inequality with mM > 0, we deduce the
following inequality that will be used in the sequel
Go Back
kf (t)k2 √
M +m
√
+ mM ≤ √
Re hf (t) , ei
mM
mM
Page 10 of 23
(2.16)
Close
Quit
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
for a.e. t ∈ [a, b] .
On the other hand
(2.17)
kf (t)k2 √
2 kf (t)k ≤ √
+ mM ,
mM
for any t ∈ [a, b] .
Utilising (2.16) and (2.17), we may conclude with the following inequality
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
M +m
kf (t)k ≤ √
Re hf (t) , ei ,
2 mM
for a.e. t ∈ [a, b] .
Applying Theorem 2.1 for the constant K :=
desired result.
S.S. Dragomir
m+M
√
2 mM
≥ 1, we deduce the
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
3.
Reverses for Orthonormal Families of Vectors
The following result for orthonormal vectors in H holds.
Theorem 3.1. Let {e1 , . . . , en } be a family of orthonormal vectors in H, ki ≥ 0,
i ∈ {1, . . . , n} and f ∈ L ([a, b] ; H) such that
ki kf (t)k ≤ Re hf (t) , ei i
(3.1)
for each i ∈ {1, . . . , n} and for a.e. t ∈ [a, b] .
Then
! 12 Z
Z b
n
b
X
2
kf (t)k dt ≤ f (t) dt
(3.2)
ki
,
i=1
a
a
where equality holds if and only if
Z b
X
Z b
n
(3.3)
f (t) dt =
kf (t)k dt
ki ei .
a
a
i=1
Rb
Proof. By Bessel’s inequality applied for a f (t) dt and the orthonormal vectors {e1 , . . . , en } , we have
Z b
2
2
n Z b
X
(3.4)
f
(t)
dt
≥
f
(t)
dt,
e
i
a
≥
i=1
n X
a
Z
Re
i=1
2
b
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 23
f (t) dt, ei
a
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
=
n Z
X
i=1
b
2
Re hf (t) , ei i dt .
a
Integrating (3.1), we get for each i ∈ {1, . . . , n}
Z b
Z b
0 ≤ ki
kf (t)k dt ≤
Re hf (t) , ei i dt,
a
a
implying
(3.5)
n Z
X
i=1
b
2 X
Z b
2
n
2
kf (t)k dt .
Re hf (t) , ei i dt ≥
ki
a
a
i=1
On making use of (3.4) and (3.5), we deduce
Z b
2
Z b
2
n
X
2
f (t) dt ≥
ki
kf (t)k dt ,
a
a
i=1
which is clearly equivalent to (3.2).
If (3.3) holds true, then
Z b
Z b
n
X
=
f
(t)
dt
kf
(t)k
dt
k
e
i i
a
a
i=1
# 12
Z b
"X
n
=
kf (t)k dt
ki2 kei k2
a
=
n
X
i=1
i=1
ki2
! 12 Z
b
kf (t)k dt,
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 23
a
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
showing that (3.2) holds with equality.
Now, suppose that there is an i0 ∈ {1, . . . , n} for which
ki0 kf (t)k < Re hf (t) , ei0 i
on a subset of nonzero Lebesgue measures enclosed in [a, b]. Then obviously
Z b
Z b
ki0
kf (t)k dt <
Re hf (t) , ei0 i dt,
a
a
and using the argument given above, we deduce
! 12 Z
Z b
n
b
X
2
.
kf (t)k dt < f
(t)
dt
ki
i=1
a
a
Therefore, if the equality holds in (3.2), we must have
ki kf (t)k = Re hf (t) , ei i
(3.6)
for each i ∈ {1, . . . , n} and a.e. t ∈ [a, b] .
Also, if the equality holds in (3.2), then we must have equality in all inequalities (3.4), this means that
Z b
n Z b
X
(3.7)
f (t) dt =
f (t) dt, ei ei
a
i=1
a
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
and
Z
(3.8)
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
Im
b
f (t) dt, ei
a
Page 14 of 23
= 0 for each i ∈ {1, . . . , n} .
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
Using (3.6) and (3.8) in (3.7), we deduce
Z
b
f (t) dt =
a
=
n
X
Re
a
Re hf (t) , ei i ei dt
a
n Z
X
i=1
Z b
b
f (t) dt, ei ei
i=1
n Z b
X
i=1
=
Z
b
kf (t)k dt ki ei
a
kf (t)k dt
=
a
n
X
ki ei ,
S.S. Dragomir
i=1
and the condition (3.3) is necessary.
This completes the proof.
Title Page
Contents
The following two corollaries are of interest.
Corollary 3.2. Let {e1 , . . . , en } be a family of orthonormal vectors in H, ρi ∈
(0, 1) , i ∈ {1, . . . , n} and f ∈ L ([a, b] ; H) such that:
(3.9)
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
kf (t) − ei k ≤ ρi for i ∈ {1, . . . , n} and a.e. t ∈ [a, b] .
JJ
J
II
I
Go Back
Close
Then we have the inequality
n−
n
X
i=1
ρ2i
Quit
! 21 Z
a
Z b
b
,
kf (t)k dt ≤ f
(t)
dt
Page 15 of 23
a
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
with equality if and only if
Z
b
Z
kf (t)k dt
f (t) dt =
a
b
a
n
X
1−
ρ2i
12
!
ei .
i=1
Proof. From the proof of Theorem 2.1, we know that (3.3) implies the inequality
q
1 − ρ2i kf (t)k ≤ Re hf (t) , ei i , i ∈ {1, . . . , n} , for a.e. t ∈ [a, b] .
Now, applying Theorem 3.1 for ki :=
desired result.
p
1 − ρ2i , i ∈ {1, . . . , n}, we deduce the
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Corollary 3.3. Let {e1 , . . . , en } be a family of orthonormal vectors in H, Mi ≥
mi > 0, i ∈ {1, . . . , n} and f ∈ L ([a, b] ; H) such that
(3.10)
Re hMi ei − f (t) , f (t) − mi ei i ≥ 0
or, equivalently,
1
M
+
m
i
i
≤ (Mi − mi )
f (t) −
e
i
2
2
for i ∈ {1, . . . , n} and a.e. t ∈ [a, b] . Then we have the reverse of the
generalised triangle inequality
" n
# 12 Z
Z b
b
X 4mi Mi
,
kf
(t)k
dt
≤
f
(t)
dt
2
(m
+
M
)
a
a
i
i
i=1
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
with equality if and only if
Z
b
Z
f (t) dt =
a
a
b
!
√
n
X
2 m i Mi
ei .
kf (t)k dt
m
+
M
i
i
i=1
Proof. From the proof of Corollary 2.3, we know (3.10) implies that
√
2 m i Mi
kf (t)k ≤ Re hf (t) , ei i , i ∈ {1, . . . , n} and a.e. t ∈ [a, b] .
m i + Mi
Now, applying Theorem 3.1 for ki :=
desired result.
√
2 mi Mi
,
mi +Mi
i ∈ {1, . . . , n} , we deduce the
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
4.
Applications for Complex-Valued Functions
Let e = α + iβ (α, β ∈ R) be a complex number with the property that |e| = 1,
i.e., α2 + β 2 = 1.
The following proposition holds.
Proposition 4.1. If f : [a, b] → C is a Lebesgue integrable function with the
property that there exists a constant K ≥ 1 such that
(4.1)
|f (t)| ≤ K [α Re f (t) + β Im f (t)]
for a.e. t ∈ [a, b] , where α, β ∈ R, α2 + β 2 = 1 are given, then we have the
following reverse of the continuous triangle inequality:
Z b
Z b
|f (t)| dt ≤ K f (t) dt .
(4.2)
a
a
The case of equality holds in (2.2) if and only if
Z b
Z b
1
f (t) dt =
(α + iβ)
|f (t)| dt.
K
a
a
The proof is obvious by Theorem 2.1, and we omit the details.
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Remark 1. If in the above Proposition 4.1 we choose α = 1, β = 0, then the
condition (4.1) for Re f (t) > 0 is equivalent to
[Re f (t)]2 + [Im f (t)]2 ≤ K 2 [Re f (t)]2
Quit
Page 18 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
or with the inequality:
|Im f (t)| √ 2
≤ K − 1.
Re f (t)
Now, if we assume that
|arg f (t)| ≤ θ,
(4.3)
π
θ ∈ 0,
,
2
then, for Re f (t) > 0,
|Im f (t)|
|tan [arg f (t)]| =
≤ tan θ,
Re f (t)
and if we choose K =
1
cos θ
> 1, then
√
K 2 − 1 = tan θ,
and by Proposition 4.1, we deduce
Z b
Z b
(4.4)
cos θ
|f (t)| dt ≤ f (t) dt ,
a
a
which is exactly the Karamata inequality (1.2) from the Introduction.
Obviously, the result from Proposition 4.1 is more comprehensive since for
other values of (α, β) ∈ R2 with α2 + β 2 = 1 we can get different sufficient
conditions for the function f such that the inequality (4.2) holds true.
A different sufficient condition in terms of complex disks is incorporated in
the following proposition.
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
Proposition 4.2. Let e = α+iβ with α2 +β 2 = 1, r ∈ (0, 1) and f : [a, b] → C
be a Lebesgue integrable function such that
(4.5)
f (t) ∈ D̄ (e, r) := {z ∈ C| |z − e| ≤ r} for a.e. t ∈ [a, b] .
Then we have the inequality
√
(4.6)
Z
1 − r2
a
b
Z b
|f (t)| dt ≤ f (t) dt .
a
The case of equality holds in (4.6) if and only if
Z b
Z b
√
2
f (t) dt = 1 − r (α + iβ)
|f (t)| dt.
a
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
a
The proof follows by Corollary 2.2 and we omit the details.
Finally, we may state the following proposition as well.
Proposition 4.3. Let e = α + iβ with α2 + β 2 = 1 and M ≥ m > 0. If
f : [a, b] → C is such that
h
i
(4.7)
Re (M e − f (t)) f (t) − me ≥ 0 for a.e. t ∈ [a, b] ,
or, equivalently,
M + m 1
(4.8)
e ≤ (M − m) for a.e. t ∈ [a, b] ,
f (t) −
2
2
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
then we have the inequality
√
Z b
Z
2 mM b
(4.9)
|f (t)| dt ≤ f (t) dt ,
M +m a
a
or, equivalently,
Z
b
(4.10) (0 ≤)
a
√
√ 2
Z b
M− m
|f (t)| dt − f (t) dt ≤
M +m
a
Z b
.
f
(t)
dt
a
The equality holds in (4.9) (or in the second part of (4.10)) if and only if
√
Z b
Z b
2 mM
(α + iβ)
|f (t)| dt.
f (t) dt =
M +m
a
a
The proof follows by Corollary 2.3 and we omit the details.
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
Remark 2. Since
M e − f (t) = M α − Re f (t) + i [M β − Im f (t)] ,
f (t) − me = Re f (t) − mα − i [Im f (t) − mβ]
JJ
J
II
I
Go Back
hence
Close
(4.11)
h
i
Re (M e − f (t)) f (t) − me
= [M α − Re f (t)] [Re f (t) − mα]
+ [M β − Im f (t)] [Im f (t) − mβ] .
Quit
Page 21 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
It is obvious that, if
(4.12)
mα ≤ Re f (t) ≤ M α for a.e. t ∈ [a, b] ,
and
(4.13)
mβ ≤ Im f (t) ≤ M β for a.e. t ∈ [a, b] ,
then, by (4.11),
h
i
Re (M e − f (t)) f (t) − me ≥ 0 for a.e. t ∈ [a, b] ,
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
and then either (4.9) or (4.12) hold true.
S.S. Dragomir
We observe that the conditions (4.12) and (4.13) are very easy to verify in
practice and may be useful in various applications where reverses of the continuous triangle inequality are required.
Title Page
Remark 3. Similar results may be stated for functions f : [a, b] → Rn or
f : [a, b] → H, with H particular instances of Hilbert spaces of significance in
applications, but we leave them to the interested reader.
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 22 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au
References
[1] J.B. DIAZ AND F.T. METCALF, A complementary triangle inequality in
Hilbert and Banach spaces, Proceedings Amer. Math. Soc., 17(1) (1966),
88–97.
[2] J. KARAMATA, Teorija i Praksa Stieltjesova Integrala (Serbo-Croatian)
(Stieltjes Integral, Theory and Practice), SANU, Posebna izdanja, 154,
Beograd, 1949.
[3] M. MARDEN, The Geometry of the Zeros of a Polynomial in a Complex
Variable, Amer. Math. Soc. Math. Surveys, 3, New York, 1949.
[4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and
New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.
[5] M. PETROVICH, Module d’une somme, L’ Ensignement Mathématique,
19 (1917), 53–56.
[6] H.S. WILF, Some applications of the inequality of arithmetic and geometric
means to polynomial equations, Proceedings Amer. Math. Soc., 14 (1963),
263–265.
Reverses of the Continuous
Triangle Inequality for Bochner
Integral of Vector-Valued
Functions in Hilbert Spaces
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 23
J. Ineq. Pure and Appl. Math. 6(2) Art. 46, 2005
http://jipam.vu.edu.au