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Young’s Integral Inequality with Upper and Lower Bounds
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Object Description
Title
Young’s
Integral
Inequality
with
Upper
and
Lower
Bounds
Author
Anderson
,
Doug
;
Noren
,
Steven
;
Perreault
,
Brent
Date Created
2011
Keywords
Young’s
inequality
;
Monotone
functions
;
Pochhammer
lower
factorial
;
Difference
equations
;
Time
scales
Abstract
Young’s
integral
inequality
is
reformulated
with
upper
and
lower
bounds
for the
remainder
. The
new
inequalities
improve
Young’s
integral
inequality
on
all
time
scales
,
such
that the
case
where
equality
holds
becomes
particularly
transparent
in this
new
presentation
. The
corresponding
results
for
difference
equations
are
given
, and
several
examples
are
included
.
We
extend
these
results
to
piecewisemonotone
functions
as
well
.
Language
English
Rights Management
Copyright owned by authors.
Research Honors
Student Lecture Series
Conference Papers & Presentations
Biography
Dr. Doug Anderson is an Associate Professor of mathematics.
Steven Noren is a mathematics major and economics minor from St. James, Minnesota.
Brent Perreault is a mathematics and physics double major from Grand Rapids, Minnesota.
Notes
This
paper
was
based
on the
results
of an
independent
research
project
,
led
by
Dr
.
Douglas
Anderson
from the
Mathematics
Department
at
Concordia
College
,
during
the
spring
semester
of
2010
. The
paper
was
presented
at the
Celebration
of
Student
Scholarship
on
April
9
,
2010
and the
Pi
Mu
Epsilon
Conference
on
April
17
,
2010
.
Submission Date
20110414
Type
Text
Description
Title
Page
1
Abstract
arXiv:1002.2463v1
[math.CA]
12
Feb
2010
YOUNG’S
INTEGRAL
INEQUALITY
WITH
UPPER
AND
LOWER
BOUNDS
DOUGLAS
R
.
ANDERSON
,
STEVEN
NOREN
, AND
BRENT
PERREAULT
Abstract
.
Young’s
integral
inequality
is
reformulated
with
upper
and
lower
bounds
for the
remainder
. The
new
inequalities
improve
Young’s
integral
inequality
on
all
time
scales
,
such
that the
case
where
equality
holds
becomes
particularly
transparent
in this
new
presentation
. The
corresponding
results
for
difference
equations
are
given
, and
several
examples
are
included
.
We
extend
these
results
to
piecewisemonotone
functions
as
well
.
1
.
introduction
In
1912
,
Young
[13]
presented
the
following
highly
intuitive
integral
inequality
,
namely
that any
real
valued
continuous
function
f
:
[0,1)
!
[0,1)
satisfying
f(0)
=
0 with
f
strictly
increasing
on
[0,1)
satisfies
ab
Z
a 0
f(t)dt
+
Z
b
0
f−1(y)dy
(1.1)
for any a,
b
2
[0,1)
, with
equality
if and
only
if
b
=
f(a)
. A
useful
consequence
of this
theorem
is
Young’s
inequality
,
ab
ap
p
+
bq
q
,
1
p
+
1
q
=
1
, with
equality
if and
only
if
ap
=
bq
, a
fact
derived
from
(1.1)
by
taking
f(t)
=
tp−1
and
q
=
p
p−1
.
Hardy
,
Littlewood
, and
P´olya
included
Young’s
inequality
in their
classic
book
[4]
, but there was
no
analytic
proof
until
Diaz
and
Metcalf
[3]
supplied
one
in
1970
.
Tolsted
[11]
showed
how to
derive
Cauchy
,
H¨older
, and
Minkowski
inequalities
in a
straightforward
way
from
(1.1)
. For
many
other
applications
and
extensions
of
Young’s
inequality
,
see
Mitrinovi´c
,
Peˇcari´c
, and
Fink
[10]
. For the
purposes
of this
paper
we
recall
some
results
that
consider
upper
bounds
for the
integrals
in
(1.1)
.
Merkle
[8]
established
the
inequality
Z
a 0
f(t)dt
+
Z
b
0
f−1(y)dy
max{af(a)
,
bf−1(b)}
,
which
has been
improved
and
reformulated
recently
by
Minguzzi
[9]
to the
inequality
0
Z
a
1
f(t)dt
+
Z
b
1
f−1(y)dy
−
ab
+
1
1
f−1(b)
−
a
(b
−
f(a))
,
(1.2)
where
the
hypotheses
of
Young’s
integral
inequality
hold
,
except
that
f(
1)
=
1
has
replaced
f(0)
=
0.
One
might
wonder
if there
is
a
discrete
version
of
(1.1)
in the
form
of a
summation
inequality
, or
more
generally
a
timescale
version
of
(1.1)
,
where
a
time
scale
,
introduced
by
Hilger
[5]
,
is
any
nonempty
closed
set
of
real
numbers
.
Wong
,
Yeh
,
Yu
, and
Hong
[12]
presented
a
version
of
Young’s
inequality
on
time
scales
T
in the
following
form
.
Using
the
standard
notation
[2]
of the
left
jump
operator
given
by
(t)
:=
sup{s
2
T
:
s
<
t}
, the
right
jump
operator
given
by
(t)
=
inf{s
2
T
:
s
>
t}
, the
compositions
f
and
f
denoted
by
f
and
f
,
respectively
, the
graininess
functions
defined
by
μ(t)
=
(t)
−t
and
2000
Mathematics
Subject
Classification
.
26D15
,
39A12
,
34N05
.
Key
words
and
phrases
.
Young’s
inequality
,
monotone
functions
,
Pochhammer
lower
factorial
,
difference
equations
,
time
scales
.
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