Journal of Inequalities in Pure and
Applied Mathematics
ČEBYŠEV’S INEQUALITY ON TIME SCALES
CHEH-CHIH YEH, FU-HSIANG WONG AND HORNG-JAAN LI
Department of Information Management
Long-Hua University of Science and Technology
KueiShan Taoyuan, 33306 Taiwan
Republic of China.
EMail: CCYeh@mail.lhu.edu.tw
Department of Mathematics
National Taipei Teacher’s College
134, Ho-Ping E. Rd. Sec. 2
Taipei 10659, Taiwan
Republic of China.
EMail: wong@tea.ntptc.edu.tw
General Education Center
Chien kuo Institute of Technology
Chang-Hua, 50050 Taiwan
Republic of China.
EMail: hjli@ckit.edu.tw
volume 6, issue 1, article 7,
2005.
Received 23 October, 2004;
accepted 21 December, 2004.
Communicated by: D. Hinton
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
204-04
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Abstract
In this paper we establish some Čebyšev’s inequalities on time scales under
suitable conditions.
2000 Mathematics Subject Classification: Primary 26B25; Secondary 26D15.
Key words: Time scales, Čebyšev’s Inequality, Delta differentiable.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
More Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References
Čebyšev’s Inequality on Time
Scales
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
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1.
Introduction
The purpose of this paper is to establish the well-known Čebyšev’s inequality on
time scales. To do this, we simply introduce the time scales calculus as follows:
In 1988, Hilger [7] introduced the time scales theory to unify continuous
and discrete analysis. A time scale T is a closed subset of the set R of the real
numbers. We assume that any time scale has the topology that it inherits from
the standard topology on R. Since a time scale may or may not be connected,
we need the concept of jump operators.
Definition 1.1. Let t ∈ T, where T is a time scale. Then the two mappings
σ, ρ : T → R
Čebyšev’s Inequality on Time
Scales
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
satisfying
Title Page
σ(t) = inf{γ > t|γ ∈ T},
ρ(t) = sup{γ < t|γ ∈ T}
are called the jump operators on T.
These jump operators classify the points {t} of a time scale T as right-dense,
right-scattered, left-dense and left-scattered according to whether σ(t) = t,
σ(t) > t, ρ(t) = t or ρ(t) < t, respectively, for t ∈ T.
Let t be the maximum element of a time scale T. If t is left-scattered, then
t is called a generate point of T. Let Tk denote the set of all non-degenerate
points of T. Throughout this paper, we suppose that
(a) T is a time scale;
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(b) an interval means the intersection of a real interval with the given time
scale;
(c) R = (−∞, ∞).
Definition 1.2. Let T be a time scale. Then the mapping f : T → R is called
rd-continuous if
(a) f is continuous at each right-dense or maximal point of T;
(b) lim− f (s) = f (t− ) exists for each left-dense point t ∈ T.
s→t
Let Crd [T, R] denote the set of all rd-continuous mappings from T to R.
Definition 1.3. Let f : T → R, t ∈ Tk . Then we say that f has the (delta)
derivative f ∆ (t) ∈ R at t if for each > 0 there exists a neighborhood U of t
such that for all s ∈ U
f (σ(t)) − f (s) − f ∆ (t)[σ(t) − s] ≤ |σ(t) − s| .
In this case, we say that f is (delta) differentiable at t.
Clearly, f ∆ is the usual derivative if T = R, and is the usual forward difference operator if T = Z (the set of all integers).
Definition 1.4. A function F : T → R is an antiderivative of f : T → R if
F ∆ (t) = f (t) for each t ∈ Tk . In this case, we define the (Cauchy) integral of
f by
Z
Čebyšev’s Inequality on Time
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and Horng-Jaan Li
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t
f (γ) ∆γ = F (t) − F (s)
Page 4 of 22
s
for all s, t ∈ T.
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It follows from Theorem 1.74 of Bohner and Peterson [3] that every rdcontinuous function has an antiderivative. For further results on time scales
calculus, we refer to [3, 9].
The purpose of this paper is to establish the well-known Čebyšev inequality
[1, 5, 6, 8, 11] on time scales. For other related results, we refer to [4, 10, 12,
13].
Čebyšev’s Inequality on Time
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Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
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2.
Main Results
We first establish some Čebyšev inequalities which generalize some results of
Audréief [1], Beesack and Pečarić [2], Dunkel [4], Fujimara [5, 6], Isayama [8],
and Winckler [13]. For other related results, we refer to the book of Mitrinovič
[10].
Theorem 2.1. Suppose that p ∈ Crd ([a, b]; [0, ∞)). Let f1 , f2 , k1 , k2 ∈ Crd ([a, b]; R)
satisfy the following two conditions:
Čebyšev’s Inequality on Time
Scales
(C1 ) f2 (x)k2 (x) > 0 on [a, b];
(C2 ) ff12 (x)
and kk12 (x)
are similarly ordered (or oppositely ordered), that is, for all
(x)
(x)
x, y ∈ [a, b],
k1 (x) k1 (y)
f1 (x) f1 (y)
−
−
≥ 0 (or ≤ 0),
f2 (x) f2 (y)
k2 (x) k2 (y)
then
(2.1)
1
2!
Z bZ
a
a
b
f1 (x) f1 (y)
p(x)p(y) f2 (x) f2 (y)
Rb
a p(x)f1 (x)k1 (x)∆x
= R
b p(x)f2 (x)k1 (x)∆x
a
k1 (x) k1 (y)
k2 (x) k2 (y)
∆x∆y
p(x)f
(x)k
(x)∆x
1
2
a
≥ 0 (or ≤ 0)
Rb
p(x)f2 (x)k2 (x)∆x Rb
a
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
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Proof. Let x, y ∈ [a, b]. Then it follows from (C1 ), (C2 ) and the identity
f1 (x) f1 (y) k1 (x) k1 (y) p(x)p(y) f2 (x) f2 (y) k2 (x) k2 (y) = p(x)p(y)f2 (x)f2 (y)k2 (x)k2 (y)
f1 (x) f1 (y)
−
f2 (x) f2 (y)
k1 (x) k1 (y)
−
k2 (x) k2 (y)
that (2.1) holds.
Čebyšev’s Inequality on Time
Scales
Remark 1. Suppose that p, f, g ∈ Crd ([a, b]; R) with p(x) ≥ 0 on [a, b]. Let
f and g be similarly ordered (or oppositely ordered). Taking f1 (x) = f (x),
k1 (x) = g(x) and f2 (x) = k2 (x) = 1, (2.1) is reduced to the generalized
Čebyšev inequality:
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
Title Page
Z
b
Z
p(x)f (x)g(x)∆x
(2.2)
a
b
Contents
p(x)∆x
a
Z
≥ (or ≤)
b
Z
p(x)f (x)∆x
a
b
p(x)g(x)∆x,
a
which generalizes a Winckler’s result in [13] if a = 0 and b = x. Let T = Z,
if a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ) are similarly ordered (or oppositely ordered), and if p = (p1 , p2 , . . . , pn ) is a nonnegative sequence, then (2.2)
is reduced to
n
X
i=1
pi
n
X
i=1
pi ai bi ≥ (or ≤)
n
X
i=1
p i ai
n
X
i=1
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p i bi .
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If T = R, then (2.2) is reduced to
Z b
Z b
Z b
Z b
p(x)f (x)g(x) dx
p(x) dx ≥ (or ≤)
p(x)f (x) dx
p(x)g(x) dx.
a
a
a
f1 (x)
,
f2 (x)
a
g1 (x)
g2 (x)
Remark 2. Taking f (x) =
g(x) =
and p(x) = f2 (x)g2 (x), inequality (2.2) is reduced to
Z b
Z b
(2.3)
f1 (x)g1 (x)∆x
f2 (x)g2 (x)∆x
a
a
Z b
Z b
≥ (or ≤)
f1 (x)g2 ∆x
f2 (x)g1 ∆x,
a
if f2 (x)g2 (x) ≥ 0 on [a, b], ff12 (x)
and
(x)
f1 (x)
ing (or one of the functions f2 (x) or
g1 (x)
g2 (x)
g1 (x)
g2 (x)
a
Theorem 2.2. Let f ∈ Crd ([a, b], [0, ∞)) be decreasing (or increasing) with
Rb
Rb
xp(x)f
(x)∆x
>
0
and
p(x)f (x)∆x > 0. Then
a
a
Rb
Rb
2
xp(x)f
(x)∆x
p(x)f 2 (x)∆x
a
a
≤ (≥) R b
.
Rb
xp(x)f (x)∆x
p(x)f (x)∆x
a
a
Proof. Clearly, for any x, y ∈ [a, b],
Z bZ b
f (x)f (y)p(x)p(y)(y − x)(f (x) − f (y)∆x∆y ≥ (≤)0,
a
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
are both increasing or both decreas-
is nonincreasing and the other nondecreasing). Here f1 , f2 , g1 , g2 ∈ Crd ([a, b], R) with f2 (x)g2 (x) 6= 0 on [a, b].
Conversely, if f1 (x) = f (x)f2 (x), g1 (x) = g(x)g2 (x) and p(x) = f2 (x)g2 (x),
then inequality (2.3) is reduced to inequality (2.2).
a
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which implies that the desired result holds.
Remark 3. Let f ∈ Crd ([a, b], (0, ∞)) and n be a positive integer. If p and g are
replaced by fp and f n respectively, then Čebyšev’s inequality (2.2) is reduced to
Z
b
Z
n
b
p(x)f (x)∆x
a
a
p(x)
∆x ≥
f (x)
b
Z
Z
b
p(x)[f (x)]n−1 ∆x,
p(x)∆x
a
a
which implies
Z
b
a
Z
p(x)
∆x
p(x)f n (x)∆x
a f (x)
Z b
Z b
Z
n−1
≥
p(x)∆x
p(x)[f (x)] ∆x
a
a
Z
a
2 Z
b
≥
Čebyšev’s Inequality on Time
Scales
2
b
p(x)
∆x
f (x)
b
p(x)[f (x)]n−2 ∆x.
p(x)∆x
a
b
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
Contents
a
provided f and f n are similarly ordered. Continuing in this way, we get
Z
a
b
p(x)
∆x
f (x)
n Z
b
n
Z
p(x)[f (x)] ∆x ≥
a
n+1
b
p(x)∆x
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,
a
which extends a result in Dunkel [4].
Remark 4. Let ν, p ∈ Crd ([a, b], [0, ∞)). If f and g are similarly ordered (or
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oppositely ordered), then it follows from Remark 1 that
Z
b
b
Z
p(t)f (ν(t))g(ν(t))∆t
a
p(t)∆t
Z b
Z b
≥ (or ≤)
p(t)f (ν(t))∆t
p(t)g(ν(t))∆t,
a
a
a
which is a generalization of a result given in Stein [12].
Remark 5. Let p, fi ∈ Crd ([a, b], R) for each i = 1, 2, . . . , n. Suppose that
f1 , f2 , . . . , fn are similarly ordered and p(x) ≥ 0 on [a, b], then it follows from
Remark 1 that
Z b
n−1 Z b
p(x)∆x
p(x)f1 (x)f2 (x) · · · fn (x)∆x
a
a
n−2 Z b
b
Z
=
p(x)∆x
a
n−2 Z
b
≥
a
b
p(x)∆x
a
Z
≥
a
Z
Z
b
p(x)f2 (x) · · · fn (x)∆x
p(x)f1 (x)∆x
a
b
p(x)f1 (x)f2 (x) · · · fn (x)∆x
p(x)∆x
a
Z
b
Z
a
b
n−3
p(x)f1 (x)∆x
p(x)∆x
a
Z b
Z b
×
p(x)f2 (x)∆x
p(x)f3 (x) · · · fn (x)∆x
a
a
Čebyšev’s Inequality on Time
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Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
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≥ ···
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b
Z
≥
b
Z
Z
p(x)f2 (x)∆x · · ·
p(x)f1 (x)∆x
p(x)fn (x)∆x ,
a
a
b
a
which is a generalization of a result in Dunkel [4].
In particular, if f1 (x) = f2 (x) = · · · = fn (x) = f (x), then
Z b
n−1 Z b
Z b
n
n
p(x)∆x
p(x)f (x)∆x ≥
p(x)f (x)∆x .
a
a
a
Theorem 2.3. If p(x), f (x) ∈ Crd ([a, b], [0, ∞)) with f (x) > 0 on [a, b] and n
is a positive integer, then
Z b
n Z b
Z b
n
p(x)
n
∆x
p(x)f (x)∆x ≥
p(x)∆x .
a f (x)
a
a
1
Proof. It follows from f (x) > 0 on [a, b] that f n (x) and f (x)
are oppositely
ordered on [a, b]. Hence by (2.2),
Z b
n
Z b
p(x)
n
p(x)f (x)∆x
∆x
a
a f (x)
Z b
n−1 Z b
Z b
p(x)
≥
p(x)∆x
∆x
p(x)f n−1 (x)∆x
f
(x)
a
a
a
Z b
2 Z b
n−2 Z b
p(x)
≥
p(x)∆x
∆x
p(x)f n−2 (x)∆x
f
(x)
a
a
a
Z b
n
≥ ··· ≥
p(x)∆x .
Čebyšev’s Inequality on Time
Scales
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
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Theorem 2.4. Let g1 , g2 , . . . , gn ∈ Crd ([a, b], <) and p, h1 , h2 , . . . , hn−1 ∈
Crd ([a, b], (0, ∞)) with gn (x) > 0 on [a, b]. If
g1 (x)g2 (x) · · · gn−1 (x)
h1 (x)h2 (x) · · · hn−1 (x)
and
hn−1 (x)
gn (x)
are similarly ordered (or oppositely ordered), then
Z
(2.4)
a
b
b
p(x)g1 (x)g2 (x) · · · gn−1 (x)
p(x)gn (x)∆x
∆x
h1 (x)h2 (x) · · · hn−2 (x)
a
Z b
Z b
p(x)g1 (x)g2 (x) · · · gn (x)
≥ (or ≤)
p(x)hn−1 (x)∆x
∆x.
a
a h1 (x)h2 (x) · · · hn−1 (x)
Z
Proof. Taking
f1 (x) =
Čebyšev’s Inequality on Time
Scales
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
Title Page
g1 (x)g2 (x) · · · gn−1 (x)
, k1 (x) = hn−1 (x),
h1 (x)h2 (x) · · · hn−1 (x)
f2 (x) = 1 and k2 (x) = gn (x)
in Theorem 2.1, (2.1) is reduced to our desired result (2.4).
The following theorem is a time scales version of Theorem 1 in Beesack and
Pečarić [2].
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Theorem 2.5. Let
f1 , f2 , . . . , fn ∈ Crd ([a, b], [0, ∞) and
Page 12 of 22
g1 , g2 , . . . , gn ∈ Crd ([a, b], (0, ∞)).
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fn
fk
If the functions f1 , fg12 , . . . , gn−1
are similarly ordered and for each pair gk−1
, gk−1
is oppositely ordered for k = 2, 3, . . . , n, then
Z
b
f2 (x)f3 (x) · · · fn (x)
∆x
g1 (x)g2 (x) · · · gn−1(x)
Rb
Rb
Rb
p(x)f
(x)∆x
p(x)f
(x)∆x
·
·
·
p(x)fn (x)∆x
1
2
≥ R ba
.
R ba
R ba
p(x)g
(x)∆x
p(x)g
(x)∆x
·
·
·
p(x)g
(x)∆x
1
2
n−1
a
a
a
p(x)f1 (x)
(2.5)
a
fn
in Remark 5, we obtain
Proof. Let f1 , f2 , . . . , fn be replaced by f1 , fg12 , . . . , gn−1
Z
n−1 Z
b
p(x)∆x
(2.6)
a
Also, since
Z
a
fk
gk−1
b
b
f2 (x)f3 (x) · · · fn (x)
p(x)f1 (x)
∆x
g1 (x)g2 (x) · · · gn−1 (x)
Z b
n Z b
Y
fk (x)
≥
p(x)f1 (x)∆x
p(x)
∆x.
g
k−1 (x)
a
a
k=2
and gk−1 are oppositely ordered, it follows from Remark 1 that
b
Z
b
Z
p(x)fk (x)∆x ≤
p(x)∆x
a
Z
p(x)gk−1 (x)∆x
a
a
b
Rb
b
p(x)
a
fk (x)
∆x.
gk−1 (x)
Čebyšev’s Inequality on Time
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Thus
Z
a
p(x)fk (x)
∆x ≥
gk−1 (x)
a
Rb
p(x)∆x a p(x)fk (x)∆x
.
Rb
p(x)g
(x)∆x
k−1
a
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Page 13 of 22
This and (2.6) imply (2.5) holds.
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3.
More Results
In this section, we generalize some results in Isayama [8].
Theorem 3.1. Let f1 , f2 , . . . , fn ∈ Crd ([a, b], (0, ∞)), k1 , k2 , . . . , kn−1 ∈
Crd ([a, b], R) and p(x) ∈ Crd ([a, b], [0, ∞)). If
f1 (x)f2 (x) · · · fi−1 (x)
k1 (x)k2 (x) · · · ki−1 (x)
and
ki−1 (x)
fi (x)
Čebyšev’s Inequality on Time
Scales
are similarly ordered (or oppositely ordered) for i = 2, . . . , n, then
Z
b
Z
b
Z
b
p(x)f2 (x)∆x · · ·
p(x)fn (x)∆x
a
Z b
Z b
≥ (or ≤)
p(x)k1 (x)∆x
p(x)k2 (x)∆x · · ·
a
a
Z b
Z b
f1 (x)f2 (x) · · · fn (x)
∆x.
···
p(x)kn−1 (x)∆x
p(x)
k1 (x)k2 (x) · · · kn−1 (x)
a
a
p(x)f1 (x)∆x
(3.1)
a
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
a
Proof. If f1 (x), k1 (x), f2 (x) and k2 (x) are replaced by f1 (x), 1, k1 (x) and
in Theorem 2.1, then we obtain
f2 (x)
k1 (x)
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Z
b
Z
p(x)f1 (x)∆x
a
a
b
p(x)f2 (x)∆x
Z b
Z b
f1 (x)f2 (x)
≥ (or ≤)
p(x)k1 (x)∆x
p(x)
∆x.
k1 (x)
a
a
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Thus the theorem holds for n = 2.
Suppose that the theorem holds for n − 1, that is
Z
b
Z
b
Z
b
p(x)f2 (x)∆x · · ·
p(x)fn−1 (x)∆x
a
a
Z b
Z b
≥ (or ≤)
p(x)k1 (x)∆x
p(x)k2 (x)∆x
a
a
Z b
Z b
f1 (x)f2 (x) · · · fn−1 (x)
p(x)kn−2 (x)∆x
p(x)
···
∆x
k1 (x)k2 (x) · · · kn−2 (x)
a
a
p(x)f1 (x)∆x
(3.2)
a
if
f1 (x)f2 (x) · · · fi−1 (x)
k1 (x)k2 (x) · · · ki−1 (x)
and
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
ki−1 (x)
fi (x)
are similarly ordered (or oppositely
ordered) for i = 2, 3, . . . , n−1. Multiplying
Rb
the both sides of (3.2) by a p(x)fn (x)∆x, we see that
Z
b
Z
b
Z
b
Z
b
p(x)f2 (x)∆x · · · p(x)fn−1 (x)∆x p(x)fn (x)∆x
a
a
Z b
Z b
≥ (or ≤) p(x)k1 (x)∆x p(x)k2 (x)∆x
a
a
Z b
Z b
Z b
f1 (x)f2 (x) · · · fn−1 (x)
∆x p(x)fn (x)∆x.
· · · p(x)kn−2 (x)∆x p(x)
k1 (x)k2 (x) · · · kn−2 (x)
a
a
a
p(x)f1 (x)∆x
(3.3)
a
a
Čebyšev’s Inequality on Time
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It follows from Theorem 2.5 that
Z b
Z b
f1 (x)f2 (x) · · · fn−1 (x)
p(x)
∆x
p(x)fn (x)∆x
k1 (x)k2 (x) · · · kn−2 (x)
a
a
Z b
Z b
f1 (x)f2 (x) · · · fn (x)
≥ (or ≤)
p(x)
∆x
p(x)kn−1 (x)∆x.
k1 (x)k2 (x) · · · kn−1 (x)
a
a
This and (3.3) imply
Z
b
Z
b
Z
b
p(x)f2 (x)∆x · · ·
p(x)fn (x)∆x
a
a
Z b
Z b
≥ (or ≤)
p(x)k1 (x)∆x
p(x)k2 (x)∆x
a
a
Z b
Z b
f1 (x)f2 (x) · · · fn (x)
···
p(x)kn−1 (x)∆x
p(x)
∆x.
k1 (x)k2 (x) · · · kn−1 (x)
a
a
p(x)f1 (x)∆x
a
By induction, we complete the proof.
Remark 6. Let kn ∈ Crd ([a, b], R). If f1 (x), f2 (x), . . . , fn (x), k1 (x), k2 (x), . . . ,
kn−1 (x) are replaced by
f1 (x)f2 (x) · · · fn (x), k1 (x)k2 (x) · · · kn (x),
...,
k1 (x)k2 (x) · · · kn (x), f1 (x)k2 (x) · · · kn (x), k1 (x)f2 (x)k3 (x) · · · kn (x),
...,
k1 (x)k2 (x) · · · kn−2 (x)fn−1 (x)kn (x)
Čebyšev’s Inequality on Time
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in Theorem 3.1, respectively, then
Z
b
p(x)f1 (x)f2 (x) · · · fn (x)∆x
(3.4)
a
Z
×
a
n−1
b
p(x)k1 (x)k2 (x) · · · kn (x)∆x
Z b
≥
p(x)f1 (x)k2 (x) · · · kn (x)∆x
a
Z b
×
p(x)k1 (x)f2 (x)k3 (x) · · · kn (x)∆x
a
Z b
···
p(x)k1 (x)k2 (x) · · · kn−1 (x)fn (x)∆x
a
if
fi (x)
ki (x)
Čebyšev’s Inequality on Time
Scales
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
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> 0 for i = 1, 2, . . . , n and k1 (x)k2 (x) · · · kn (x) > 0 on [a, b].
Contents
Remark 7. Letting f1 (x) = f2 (x) = · · · = fn (x) = f (x) and k1 (x) = k2 (x) =
1
· · · = kn (x) = k n−1 (x) in (3.4) with k(x) > 0 on [a, b], we obtain the Hölder
inequality:
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Z
(3.5)
a
b
p(x)f n (x)∆x
Z
b
n
n−1
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p(x)k n−1 (x)∆x
a
Z
≥
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n
b
p(x)f (x)k(x)∆x
.
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a
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Remark 8. Let p, f, g ∈ Crd ([a, b], [0, ∞)). Taking
f1 (x) = f n (x)g(x),
f2 (x) = f3 (x) = · · · = fn (x) = g(x) and
k1 (x) = k2 (x) = · · · = kn−1 (x) = f (x)g(x),
(3.1) is reduced to Jensen’s inequality:
Z
b
Z
n
p(x)f (x)g(x)∆x
(3.6)
a
n−1
b
Čebyšev’s Inequality on Time
Scales
p(x)g(x)∆x
a
Z
≥
n
b
p(x)f (x)g(x)∆x
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
.
a
1
Remark 9. Taking k1 (x) = k2 (x) = · · · = kn−1 (x) = (f1 (x)f2 (x) · · · fn (x)) n ,
(3.1) is reduced to
Z
b
Z
p(x)f1 (x)∆x
(3.7)
a
a
b
Z
b
p(x)f2 (x)∆x · · ·
p(x)fn (x)∆x
a
Z b
n
1
n
≥
p(x) (f1 (x)f2 (x) · · · fn (x)) ∆x
a
if fi (x) > 0 on [a, b] for each i = 1, 2, . . . , n and
(i = 1, 2, . . . , n) are similarly ordered.
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1
1
[f (x)f2 (x) · · · fn (x)] n
fi (x) 1
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Remark 10 (see also Remark 5). Taking k1 (x) = k2 (x) = · · · = kn−1 (x) = 1,
(3.1) is reduced to Čebyšev’s inequality:
b
Z
Z
b
p(x)f2 (x)∆x · · ·
p(x)f1 (x)∆x
(3.8)
Z
a
p(x)fn (x)∆x
a
Z
≤
n−1 Z
b
b
a
b
p(x)f1 (x)f2 (x) · · · fn (x)∆x
p(x)∆x
a
a
if fi (x) (i = 1, 2, . . . , n) are similarly ordered and fi (x) ≥ 0 (i = 1, 2, . . . , n).
Remark 11. Taking f1 (x) = f2 (x) = · · · = fn (x) = 1, then (3.1) is reduced to
Z
n
b
Z
≤
p(x)∆x
a
b
Z
Čebyšev’s Inequality on Time
Scales
Cheh-Chih Yeh, Fu-Hsiang Wong,
and Horng-Jaan Li
b
p(x)k1 (x)∆x
p(x)k2 (x)∆x
a
a
Z b
Z b
p(x)
∆x
···
p(x)kn−1 (x)∆x
a
a k1 (x)k2 (x) · · · kn−1 (x)
if ki (x) > 0 are similarly ordered for i = 1, 2, . . . , n−1. Thus, if f1 (x), . . . , fn (x)
are similarly ordered and fi (x) > 0 on [a, b] (i = 1, 2, . . . , n), then
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R
(3.9) R b
b
a
p(x)∆x
n+1
p(x)
∆x
a f1 (x)f2 (x)···fn (x)
Z b
≤
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Z
Z
p(x)f2 (x)∆x · · ·
p(x)f1 (x)∆x
a
b
a
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b
p(x)fn (x)∆x.
a
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It follows from (3.8) and (3.9) that
R
n+1
b
Z b
n−1 Z b
p(x)∆x
a
≤
p(x)∆x
p(x)f1 (x)f2 (x) · · · fn (x)∆x
Rb
p(x)
∆x
a
a
a f1 (x)f2 (x)···fn (x)
if f1 (x), . . . , fn (x) are similarly ordered.
Remark 12. Let k1 (x) = k2 (x) = · · · = kn (x) = 1. If fi (x) is replaced by
1
[f1 (x)f2 (x) · · · fn (x)] n
,
fi (x)
n = 1, 2, . . . , n,
then (3.1) is reduced to
p
Z b
n
n Z b
n
Y
f1 (x)f2 (x) · · · fn (x)
p(x)
∆x ≤
p(x)∆x
fi (x)
a
i=1 a
√
n
f1 (x)f2 (x)···fn (x)
if
(i = 1, 2, . . . , n) are similarly ordered.
fi (x)
Remark 13. Let f1 , f2 , . . . , fn ; k1 , k2 , . . . , kn−1 be replaced by f1 f2 , f3 f4 , . . . ,
f2n−1 f2n ; f2 f3 , f4 f5 , . . . , f2n−2 f2n−1 , respectively. Then (3.1) is reduced to
Z b
Z b
Z b
p(x)f1 (x)f2 (x)∆x
p(x)f3 (x)f4 (x)∆x · · ·
p(x)f2n−1 (x)f2n (x)∆x
a
a
a
Z b
Z b
Z b
≥
p(x)f2 (x)f3 (x)∆x
p(x)f4 (x)f5 (x)∆x · · ·
p(x)f2n−2 (x)f2n−1 (x)∆x
a
if
fi (x)
fi+1 (x)
a
a
Čebyšev’s Inequality on Time
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(i = 1, 2, . . . , 2n − 1) are similarly ordered.
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References
[1] K.A. ANDRÉIEF, N’skol’ko slov’ po povodu teorem’ P. L. Čebyševa i V.
G. Imšeneckogo ob’ opred’lennyh’ integralah’ ot’ proizvedenyja funkciǐ,
Soobščeuija i Protokoly Zasedanii Matematičeskogo Obščestva Pri Imperatorskom Har’kovskom Universitete, 1883, 110–123.
[2] P.R. BEESACK AND J.E. PEČARIĆ, Integral inequalities of Čebyšev
type, J. Math. Anal. Appl., 111 (1985), 643–695.
[3] M. BOHNER AND A. PETERSON, “Dynamic Equations on Time Scales",
Birkhäuser, Boston/Basel/Berlin, 2001.
[4] O. DUNKEL, Integral inequalities with applications to the calculus of variants, Amer. Math. Monthly, 31 (1924), 326–337.
[5] M. FUJIWARA, Ein von Brunn vermuteter Satz über konvexe Flächen
und eine Verallgemeinerung der Schwarzschen und der Tchebycheffschen
Ungleichungen für bestimmte Integrale, Tôhoku Math. J., 13 (1918), 228–
235.
[6] M. FUJIWARA, Über eine Ungleichung für bestimmte Integrale, Tôhoku
Math. J,. 15 (1919), 285–288.
[7] S. HILGER, Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math., 18 (1900), 18–56.
[8] S. ISAYAMA, Extension of the known integral inequalities, Tôhoku Math.
J., 26 (1925/26), 238–246.
Čebyšev’s Inequality on Time
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and Horng-Jaan Li
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[9] V. LAKSHMIKANTHAM, S. SIVASUNDARAM AND B. KAYMAKCALAN, Dynamic Systems on Measure Chains, Klumer Academic Publishers, Dordrecht/Boston/London, 1996.
[10] D.S. MITRINOVIĆ, Analysis Inequalities,
York/Heidelberg/Berlin, 1970.
Springer-Verlay,
New
[11] S. NARUMI, Note on an inequality for definite integrals, Tôhoku Math. J.,
27 (1926), 258–266.
[12] S.K. STEIN, An inequality in two monotonic functions, Amer. Math.
Monthly, 83 (1976), 469–471.
[13] A. WINCKLER, Allgemeine Sätze zur Theorie der unregelmässigen
Beobachtungsfehler, Sigzungsberichte der Wiener Akademie, 53 (1866),
6–41.
Čebyšev’s Inequality on Time
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