GENERALIZATIONS OF SOME NEW ČEBYŠEV
TYPE INEQUALITIES
Čebyšev Type Inequalities
ZHENG LIU
Institute of Applied Mathematics, School of Science
University of Science and Technology Liaoning
Anshan 114051, Liaoning, China
EMail: lewzheng@163.net
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
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Received:
17 August, 2006
Accepted:
02 January, 2007
Communicated by:
JJ
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N.S. Barnett
J
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2000 AMS Sub. Class.:
26D15.
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Key words:
Cebyšev type inequalities, Absolutely continuous functions, Cauchy-Schwarz inequality for double integrals, Lp spaces, Hölder’s integral inequality.
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Abstract:
We provide generalizations of some recently published Čebyšev type inequalities.
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Acknowledgements:
The author wishes to thank the editor for his help with the final presentation of
this paper.
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Contents
1
Introduction
3
2
Statement of Results
6
3
Proof of Theorem 2.1
8
Čebyšev Type Inequalities
4
Proof of Theorem 2.2
11
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
5
Proof of Theorem 2.3
13
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1.
Introduction
In a recent paper [1], B.G. Pachpatte proved the following Čebyšhev type inequalities:
Theorem 1.1. Let f, g : [a, b] → R be absolutely continuous functions on [a, b] with
f 0 , g 0 ∈ L2 [a, b], then,
(1.1)
(1.2)
12
(b − a)2
1
0 2
2
|P (F, G, f, g)| ≤
kf k2 − ([f ; a, b])
12
b−a
12
1
0 2
2
,
×
kg k2 − ([g; a, b])
b−a
12
1
(b − a)2
0 2
2
|P (A, B, f, g)| ≤
kf k2 − ([f ; a, b])
12
b−a
12
1
0 2
2
×
kg k2 − ([g; a, b])
,
b−a
where
(1.3)
(1.4)
Čebyšev Type Inequalities
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
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Z b
Z b
1
P (α, β, f, g) = αβ −
α
g(t) dt + β
f (t) dt
b−a
a
a
Z b
1
1
+
f (t) dt
g(t) dt ,
b−a a
b−a
[f ; a, b] =
f (b) − f (a)
,
b−a
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f (a) + f (b)
F =
,
2
and
g(a) + g(b)
G=
,
2
Z
A=f
b
kf k2 :=
a+b
2
,
B=g
a+b
2
,
12
f 2 (t) dt .
a
Theorem 1.2. Let f, g : [a, b] → R be differentiable functions so that f 0 , g 0 are
absolutely continuous on [a, b], then,
4
(1.5)
|P (F , G, f, g)| ≤
(b − a)
kf 00 − [f 0 ; a, b]k∞ kg 00 − [g 0 ; a, b]k∞ ,
144
where
Čebyšev Type Inequalities
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
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f (a) + f (b) (b − a)2 0
−
[f ; a, b],
2
12
g(a) + g(b) (b − a)2 0
G=
−
[g ; a, b],
2
12
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F =
P (α, β, f, g) and [f ; a, b] are as defined in (1.3) and (1.4), and
kf k∞ = sup |f (t)| < ∞.
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t∈[a,b]
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In [2], B.G. Pachpatte presented an additional Čebyšev type inequality given in
Theorem 1.3 below.
Theorem 1.3. Let f, g : [a, b] → R be absolutely continuous functions whose derivatives f 0 , g 0 ∈ Lp [a, b], p > 1, then we have,
(1.6)
|P (C, D, f, g)| ≤
2
1
M q kf 0 kp kg 0 kp ,
2
(b − a)
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where P (α, β, f, g) is as defined in (1.3),
1 f (a) + f (b)
a+b
C=
+ 2f
,
3
2
2
1 g(a) + g(b)
a+b
D=
+ 2g
,
3
2
2
Čebyšev Type Inequalities
M=
(1.7)
with
1
p
+
1
q
(2q+1 + 1)(b − a)q+1
3(q + 1)6q
= 1, and
Z
kf kp =
b
p1
|f (t)|p dt
< ∞.
a
In this paper, we provide some generalizations of the above three theorems.
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
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2.
Statement of Results
We use the following notation to simplify the detail of presentation. For suitable
functions f, g : [a, b] → R and real number θ ∈ [0, 1] we set,
a+b
θ
Γθ = [f (a) + f (b)] + (1 − θ)f
,
2
2
a+b
θ
∆θ = [g(a) + g(b)] + (1 − θ)g
,
2
2
(1 − 3θ)(b − a)2 0
Γθ = Γθ +
[f , a, b],
24
(1 − 3θ)(b − a)2 0
∆θ = ∆θ +
[f , a, b],
24
where [f ; a, b] is as defined in (1.4).
We also use P (α, β, f, g) as defined in (1.3), where α and β are real constants.
The results are stated as Theorems 2.1, 2.2 and 2.3.
Theorem 2.1. Let the assumptions of Theorem 1.1 hold, then for any θ ∈ [0, 1],
(b − a)2 3
(2.1) |P (Γθ , ∆θ , f, g)| ≤
[θ + (1 − θ)3 ]
12
12 12
1
1
×
kf 0 k22 − ([f ; a, b])2
kg 0 k22 − ([g; a, b])2 .
b−a
b−a
Theorem 2.2. Let the assumptions of Theorem 1.2 hold, then for any θ ∈ [0, 1],
(2.2)
|P (Γθ , ∆θ , f, g)| ≤ (b − a)4 I 2 (θ)kf 00 − [f 0 ; a, b]k∞ kg 00 − [g 0 ; a, b]k∞ ,
Čebyšev Type Inequalities
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
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where
(
I(θ) =
(2.3)
θ3
3
− 8θ +
1
(θ
8
− 13 ),
1
,
24
0 ≤ θ ≤ 12 ,
1
2
< θ ≤ 1.
Theorem 2.3. Let the assumptions of Theorem 1.3 hold, then for any θ ∈ [0, 1],
|P (Γθ , ∆θ , f, g)| ≤
(2.4)
2
1
q
0
0
M
θ kf kp kg kp ,
2
(b − a)
Čebyšev Type Inequalities
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
where
θq+1 + (1 − θ)q+1
Mθ =
(b − a)q+1 ,
q
(q + 1)2
(2.5)
and
1
p
+
1
q
= 1.
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3.
Proof of Theorem 2.1
Define the function,
(
K(θ, t) =
(3.1)
t − (a + θ b−a
),
2
t ∈ [a, a+b
],
2
t − (b − θ b−a
),
2
t ∈ ( a+b
, b],
2
and we obtain the following identities:
Z b
1
(3.2)
Γθ −
f (t) dt = O(f ; a, b; θ),
b−a a
Čebyšev Type Inequalities
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
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∆θ −
(3.3)
1
b−a
b
Z
a
where
O(f ; a, b; θ) =
1
2(b − a)2
Z bZ
a
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g(t) dt = O(g; a, b; θ),
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b
(f 0 (t) − f 0 (s))(k(θ, t) − k(θ, s) dt ds.
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a
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Multiplying the left sides and right sides of (3.2) and (3.3) we get,
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(3.4)
P (Γθ , ∆θ , f, g) = O(f ; a, b; θ)O(g; a, b; θ).
From (3.4),
(3.5)
|P (Γθ , ∆θ , f, g)| = |O(f ; a, b; θ)||O(g; a, b; θ)|.
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Using the Cauchy-Schwarz inequality for double integrals,
Z bZ b
1
(3.6) |O(f ; a, b; θ)| ≤
|f 0 (t) − f 0 (s)||k(θ, t) − k(θ, s)| dt ds
2(b − a)2 a a
12
Z bZ b
1
0
0
2
≤
(f (t) − f (s)) dt ds
2(b − a)2 a a
12
Z bZ b
1
×
(k(θ, t) − k(θ, s))2 dt ds .
2(b − a)2 a a
By simple computation,
Z bZ b
1
(f 0 (t) − f 0 (s))2 dt ds
(3.7)
2(b − a)2 a a
2
Z b
Z b
1
1
0
2
0
=
(f (t)) dt −
f (t)dt ,
b−a a
b−a a
and
(3.8)
1
2(b − a)2
Z bZ
a
b
(k(θ, t) − K(θ, s))2 dt ds =
a
(b − a)2 3
[θ + (1 − θ)3 ].
12
Using (3.7), (3.8) in (3.6),
|O(f ; a, b; θ)| ≤
1
b−a 3
1
√ [θ + (1 − θ)3 ] 2
kf 0 k22 − ([f ; a, b])2
b−a
2 3
21
.
Similarly,
(3.10)
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(3.9)
Čebyšev Type Inequalities
12
1
b−a 3
1
|O(g; a, b; θ)| ≤ √ [θ + (1 − θ)3 ] 2
kg 0 k22 − ([g; a, b])2 .
b−a
2 3
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Using (3.9) and (3.10) in (3.5), (2.1) follows.
Remark 1. If θ = 1 and θ = 0 in (2.1), the inequalities (1.1) and (1.2) are recaptured.
Thus Theorem 2.1 may be regarded as a generalization of Theorem 1.1.
Čebyšev Type Inequalities
Zheng Liu
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4.
Proof of Theorem 2.2
Define the function
(
L(θ, t) =
1
(t
2
− a)[t − (1 − θ)a − θb],
t ∈ [a, a+b
],
2
1
(t
2
− b)[t − θa − (1 − θ)b],
t ∈ ( a+b
, b].
2
It is not difficult to find the following identities:
Z b
1
(4.1)
f (t) dt − Γθ = Q(f 0 , f 00 ; a, b),
b−a a
1
b−a
(4.2)
Z
b
g(t) dt − ∆θ = Q(g 0 , g 00 ; a, b),
Čebyšev Type Inequalities
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
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a
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where
b
1
L(θ, t){f 00 (t) − [f 0 ; a, b]} dt.
b−a a
Multiplying the left sides and right sides of (4.1) and (4.2), we get,
Q(f 0 , f 00 ; a, b) =
Z
P (Γθ , ∆θ , f, g) = Q(f 0 , f 00 ; a, b)Q(g 0 , g 00 ; a, b).
(4.3)
|P (Γθ , ∆θ , f, g)| = |Q(f 0 , f 00 ; a, b)||Q(g 0 , g 00 ; a, b)|.
By simple computation, we have,
(4.5)
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From (4.3),
(4.4)
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b
1
|L(θ, t)||[f 00 (t) − [f 0 ; a, b]| dt
b−a a
Z b
1
00
0
≤
kf (t) − [f ; a, b]k∞
|L(θ, t)| dt,
b−a
a
|Q(f 0 , f 00 ; a, b)| ≤
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Z
and similarly,
(4.6)
1
|Q(f , f ; a, b)| ≤
kf 00 (t) − [f 0 ; a, b]k∞
b−a
0
00
b
Z
|L(θ, t)| dt,
a
where
Z
(
b
3
|L(θ, t)| dt = (b − a) ×
(4.7)
a
θ3
3
− 8θ +
1
(θ
8
− 13 ),
1
,
24
0 ≤ θ ≤ 12 ,
1
2
Čebyšev Type Inequalities
Zheng Liu
< θ ≤ 1.
Consequently, the inequalities (2.2) and (2.3) follow from (4.4) – (4.7).
Remark 2. If θ = 1 in (2.2) with (2.3), the inequality (1.5) is recaptured. Thus
Theorem 2.2 may be regarded as a generalization of Theorem 1.2.
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5.
Proof of Theorem 2.3
From (3.1), we can also find the following identities:
Z b
Z b
1
1
(5.1)
Γθ −
f (t) dt =
K(θ, t)f 0 (t) dt,
b−a a
b−a a
(5.2)
1
∆θ −
b−a
Z
a
b
1
g(t) dt =
b−a
b
Z
Čebyšev Type Inequalities
K(θ, t)g 0 (t) dt.
a
Multiplying the left sides and right sides of (5.1) and (5.2) we get,
Z b
Z b
1
0
0
(5.3) P (Γθ , ∆θ , f, g) =
k(θ, t)f (t) dt
k(θ, t)g (t) dt .
(b − a)2
a
a
From (5.3) and using the properties of modulus and Hölder’s integral inequality, we
have,
(5.4)
|P (Γθ ,∆θ , f, g)|
Z b
b
1
0
0
≤
|k(θ, t)||f (t)| dt
|k(θ, t)||g (t)| dt
(b − a)2
a
a
" Z
1q Z b
p1 #
b
1
≤
|k(θ, t)|q dt
|f 0 |p dt
(b − a)2
a
a
" Z
1q Z b
p1 #
b
×
|k(θ, t)|q dt
|g 0 |p dt
Z
a
=
1
(b − a)2
a
Z
a
2q
b
q
|k(θ, t)| dt kf 0 kp kg 0 kp .
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vol. 8, iss. 1, art. 13, 2007
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A simple computation gives,
Z b
(5.5)
|k(θ, t)|q dt
a
a+b
2
q
q
Z b b
−
a
b
−
a
t − a + θ
dt +
t − b − θ
dt
=
a+b
2
2
a
2
q
q
Z a+b Z a+θ b−a 2
2
b−a
b−a
dt
=
a+θ
− t dt +
t−a−θ
2
2
a+θ b−a
a
2
q
q
Z b−θ b−a Z b
2
b−a
b−a
dt
+
b−θ
− t dt +
t−b+θ
a+b
2
2
b−θ b−a
2
2
" #
q+1
q+1
θ
1
−
θ
2
(b − a)q+1 +
(b − a)q+1
=
q+1
2
2
Z
=
θ
q+1
Čebyšev Type Inequalities
Zheng Liu
vol. 8, iss. 1, art. 13, 2007
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q+1
+ (1 − θ)
(q + 1)2q
(b − a)q+1 = Mθ .
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Consequently, the inequality (2.4) with (2.5) follow from (5.4) and (5.5).
Remark 3. If we take θ = 13 in (2.4) with (2.5), we recapture the inequality (1.6)
with (1.7). Thus Theorem 2.3 may be regarded as a generalization of Theorem 1.3.
Remark 4. If we take p = 2 in Theorem 2.3, and replace f (t) and g(t) by f (t) −
[f ; a, b]t and g(t) − [g; a, b]t in (2.4), respectively, then inequality (2.1) is recaptured.
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References
[1] B.G. PACHPATTE, New Čebyšev type inequalities via trapezoidal-like rules,
J. Inequal. Pure and Appl. Math., 7(1) (2006), Art. 31. [ONLINE: http://
jipam.vu.edu.au/article.php?sid=637].
[2] B.G. PACHPATTE, On Čebyšev type inequalities involving functions whose
derivatives belong to Lp spaces, J. Inequal. Pure and Appl. Math., 7(2) (2006),
Art. 58. [ONLINE: http://jipam.vu.edu.au/article.php?sid=
675].
Čebyšev Type Inequalities
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vol. 8, iss. 1, art. 13, 2007
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