Journal of Inequalities in Pure and
Applied Mathematics
A UNIFIED TREATMENT OF SOME SHARP INEQUALITIES
ZHENG LIU
Institute of Applied Mathematics, Faculty of Science
Anshan University of Science and Technology
Anshan 114044, Liaoning, China
EMail: lewzheng@163.net
volume 7, issue 5, article 172,
2006.
Received 24 July, 2006;
accepted 11 December, 2006.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
A generalization of some recent sharp inequalities by N. Ujević is established.
Applications in numerical integration are also considered.
2000 Mathematics Subject Classification: Primary 26D15
Key words: Quadrature formula, Sharp error bounds, Generalization, Numerical integration.
Contents
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Applications in Numerical Integration . . . . . . . . . . . . . . . . . . . 13
References
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1.
Introduction
In [1] we can find a generalization of the pre-Grüss inequality as:
Lemma 1.1. Let f, g, Ψ ∈ L2 (a, b). Then we have
SΨ (f, g)2 ≤ SΨ (f, f )SΨ (g, g),
(1.1)
where
Z
(1.2) SΨ (f, g) =
a
b
1
f (t)g(t) dt −
b−a
−
Z
b
a
1
kΨk22
Z
b
f (t) dt
g(t) dt
a
Z b
Z b
f (t)Ψ(t) dt
g(t)Ψ(t) dt
a
a
and Ψ satisfies
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
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Z
b
Ψ(t) dt = 0,
(1.3)
a
while as usual, k · k2 is the norm in L2 (a, b). i.e.,
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kΨk22 =
Z
b
Ψ2 (t) dt.
a
Using the above inequality, Ujević in [1] obtained the following interesting
results:
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Theorem 1.2. Let f : [a, b] → R be an absolutely continuous function whose
derivative f 0 ∈ L2 (a, b). Then
Z b
(b − a) 23
a
+
b
f
√ C1
(1.4)
(b − a) −
f (t) dt ≤
2
2 3
a
where
(1.5)
C1 =
[f (b) − f (a)]2
kf 0 k22 −
− [Q(f ; a, b)]2
b−a
12
A Unified Treatment of Some
Sharp Inequalities
and
Z b
a+b
2
3
(1.6) Q(f ; a, b) = √
f (a) + f
+ f (b) −
f (t) dt .
2
b−a a
b−a
Theorem 1.3. Let the assumptions of Theorem 1.2 hold. Then
Z b
f (a) + f (b)
(b − a) 32
√ C2 ,
(1.7)
(b − a) −
f (t) dt ≤
2
2 3
a
where
(1.8)
C2 =
kf 0 k22
[f (b) − f (a)]2
−
− [P (f ; a, b)]2
b−a
12
and
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(1.9) P (f ; a, b)
Z b
1
6
a+b
=√
f (a) + 4f
+ f (b) −
f (t) dt .
2
b−a a
b−a
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Theorem 1.4. Let the assumptions of Theorem 1.2 hold. Then
Z b
f (a) + 2f a+b + f (b)
(b − a) 32
2
√ C3 ,
(1.10)
(b − a) −
f (t) dt ≤
4
4 3
a
where
(1.11)
[f (b) − f (a)]2
C3 = kf 0 k22 −
b−a
2 ) 12
1
a+b
−
f (a) − 2f
+ f (b)
b−a
2
(
2
a+b
2
0 2
= kf k2 −
f
− f (a)
b−a
2
2 ) 12
2
a+b
−
f (b) − f
.
b−a
2
In [2], Ujević further proved that the above all inequalities are sharp.
In this paper, we will derive a new sharp inequality with a parameter for
absolutely continuous functions with derivatives belonging to L2 (a, b), which
not only provides a unified treatment of all the above sharp inequalities, but also
gives some other interesting results as special cases. Applications in numerical
integration are also considered.
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
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2.
Main Results
Theorem 2.1. Let the assumptions of Theorem 1.2 hold. Then for any θ ∈ [0, 1]
we have
Z b
a+b
f (a) + f (b)
(2.1) (b − a) (1 − θ)f
+θ
−
f (t) dt
2
2
a
3
√
(b − a) 2
√
1 − 3θ + 3θ2 C(θ),
≤
2 3
where
(2.2)
C(θ) =
kf 0 k22
[f (b) − f (a)]2
−
− [N (f ; a, b; θ)]2
b−a
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
12
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and
(2.3) N (f ; a, b; θ) = p
Contents
2
(1 − 3θ + 3θ2 )(b − a)
Z
a+b
f (a) + f (b) 3 − 6θ b
× (1 − 3θ)f
+ (2 − 3θ)
−
f (t) dt .
2
2
b−a a
The inequality (2.1) with (2.2) and (2.3) is sharp in the sense that the constant
1
√
cannot be replaced by a smaller one.
2 3
Proof. Let us define the functions
(
t − a,
p(t) =
t − b,
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,
t ∈ a, a+b
2
t ∈ a+b
,b ,
2
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and
t − a + θ b−a
,
2
t − b − θ b−a
,
2
(
Ψ(t) =
t ∈ a, a+b
,
2
t ∈ a+b
,b ,
2
where θ ∈ [0, 1].
It is not difficult to verify that
Z b
Z b
(2.4)
p(t) dt =
Ψ(t) dt = 0.
a
a
i.e., Ψ satisfies the condition (1.3).
We also have
Z b
(b − a)3
2
(2.5)
kpk2 =
p2 (t) dt =
12
a
and
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
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kΨk22 =
(2.6)
Z
b
Ψ2 (t) dt =
a
We now calculate
Z b
Z
p(t)Ψ(t) dt =
(2.7)
a
a+b
2
(b − a)3
(1 − 3θ + 3θ2 ).
12
b−a
(t − a) t − a − θ
dt
2
a
Z b
b−a
+
(t − b) t − b + θ
dt
a+b
2
2
1
θ
=
−
(b − a)3 .
12 8
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Integrating by parts, we have
Z b
Z b
a+b
0
(b − a) −
(2.8)
f (t)p(t) dt = f
f (t) dt
2
a
a
and
Z
(2.9)
b
f 0 (t)Ψ(t) dt
a
a+b
2
Z b b−a
b−a
0
f 0 (t) dt
=
t−a−θ
f (t) dt +
t−b+θ
a+b
2
2
a
2
Z b
a+b
f (a) + f (b)
= (b − a) (1 − θ)f
+θ
−
f (t) dt.
2
2
a
Z
From (2.4), (2.6) – (2.9) and (1.2) we get
(2.10) SΨ (f 0 , p)
Z b
Z b
Z b
1
0
0
=
f (t)p(t) dt −
f (t) dt
p(t) dt
b−a a
a
a
Z b
Z b
1
0
f (t)Ψ(t) dt
p(t)Ψ(t) dt
−
kΨk22 a
a
Z b
a+b
2 − 3θ
=f
(b − a) −
f (t) dt −
2
2(1 − 3θ + 3θ2 )
a
Z b
a+b
f (a) + f (b)
× (b − a) (1 − θ)f
+θ
−
f (t) dt
2
2
a
A Unified Treatment of Some
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Zheng Liu
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θ
a+b
=
(b − a) (1 − 3θ)f
2(1 − 3θ + 3θ2 )
2
Z b
f (a) + f (b)
+ (2 − 3θ)
− (3 − 6θ)
f (t) dt .
2
a
From (2.4) – (2.7) and (1.2) we also have
Z b
2
Z b
2
1
1
2
(2.11) SΨ (p, p) = kpk2 −
p(t) dt −
p(t)Ψ(t) dt
b−a
kΨk22
a
a
θ2 (b − a)3
=
16(1 − 3θ + 3θ2 )
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
and
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(2.12)
0
0
SΨ (f , f )
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Z
b
2
0
f (t) dt −
Z
b
2
0
f (t)Ψ(t) dt
1
1
b−a
kΨk22
a
a
2
12
[f
(b)
−
f
(a)]
= kf 0 k22 −
−
b−a
(1 − 3θ + 3θ2 )(b − a)
2
Z b
a+b
f (a) + f (b)
1
× (1 − θ)f
+θ
−
f (t) dt
2
2
b−a a
= kf 0 k22 −
Thus from (2.10) – (2.12) and (1.1) we can easily get
a
+
b
f
(a)
+
f
(b)
(2.13) (b − a) (1 − 3θ)f
+ (2 − 3θ)
2
2
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2
(1 − 3θ + 3θ2 )(b − a)3
−(3 − 6θ)
f (t) dt ≤
4
a
2
12
[f (b) − f (a)]
× kf 0 k22 −
−
b−a
(1 − 3θ + 3θ2 )(b − a)
2 )
Z b
a+b
f (a) + f (b)
1
× (1 − θ)f
.
+θ
−
f (t) dt
2
2
b−a a
Z
b
It is equivalent to
2
Z b
f (a) + f (b)
a+b
1
(2.14) 3(b − a) (1 − θ)f
+θ
−
f (t) dt
2
2
b−a a
1 − 3θ + 3θ2
[f (b) − f (a)]2
4
3
≤
(b − a) kf 0 k22 −
−
4
b−a
(1 − 3θ + 3θ2 )(b − a)
2 )
Z b
a
+
b
f
(a)
+
f
(b)
3
−
6θ
× (1 − 3θ)f
+ (2 − 3θ)
−
f (t) dt .
2
2
b−a a
A Unified Treatment of Some
Sharp Inequalities
2
Consequently, inequality (2.1) with (2.2) and (2.3) follow from (2.14).
In order to prove that the inequality (2.1) with (2.2) and (2.3) is sharp for any
θ ∈ [0, 1], we define the function

 21 t2 − 2θ t,
t ∈ 0, 12 ,
(2.15)
f (t) =
 1 t2 − 1 − θ t + 1−θ , t ∈ 1 , 1
2
2
2
2
The function given in (2.15) is absolutely continuous since it is a continuous
piecewise polynomial function.
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We now suppose that (2.1) holds with a constant C > 0 as
(2.16)
Z b
(b − a) (1 − θ)f a + b + θ f (a) + f (b) −
f (t) dt
2
2
a
3√
2
≤ C(b − a) 1 − 3θ + 3θ2 C(θ),
where C(θ) is as defined in (2.2) and (2.3).
Choosing a = 0, b = 1, and f defined in (2.15), we get
Z 1
1
θ
f (t) dt =
− ,
24 8
0
1
1 θ
f (0) = f (1) = 0,
f
= − ,
2
8 4
Z 1
1 − 3θ + 3θ2
(f 0 (t))2 dt =
12
0
and
N (f ; a, b; θ) = 0
1 − 3θ + 3θ2
L.H.S.(2.16) =
.
12
R.H.S.(2.16) =
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We also find that the right-hand side is
(2.18)
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such that the left-hand side becomes
(2.17)
A Unified Treatment of Some
Sharp Inequalities
C(1 − 3θ + 3θ2 )
√
.
2 3
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From (2.16) – (2.18), we find that C ≥
best possible in (2.1).
1
√
,
2 3
proving that the constant
1
√
2 3
is the
Remark 1. If we take θ = 0, θ = 1 and θ = 12 in (2.1) with (2.2) and (2.3),
we recapture the sharp midpoint type inequality (1.4) with (1.5) and (1.6), the
sharp trapezoid type inequality (1.7) with (1.8) and (1.9) and the sharp averaged midpoint-trapezoid type inequality (1.10) with (1.11), respectively. Thus
Theorem 2.1 may be regarded as a generalization of Theorem 1.2, Theorem 1.3
and Theorem 1.4.
Remark 2. If we take θ =
1
,
3
we get a sharp Simpson type inequality as
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
(2.19)
Z b
(b − a) 32
b − a
a+b
C4 ,
f (a) + 4f
+ f (b) −
f (t) dt ≤
6
6
2
a
where
(2.20)
Contents
12
2
[f
(b)
−
f
(a)]
C4 = kf 0 k22 −
− [R(f ; a, b)]2
b−a
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and
1
R(f ; a, b) = N f ; a, b;
3
√ Z b
2 3 f (a) + f (b)
1
=√
−
f (t) dt .
2
b−a a
b−a
(2.21)
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3.
Applications in Numerical Integration
We restrict further considerations to the averaged midpoint-trapezoid quadrature rule. We also emphasize that similar considerations may be made for all
the quadrature rules considered in the previous section.
Theorem 3.1. Let π = {x0 = a < x1 < · · · < xn = b} be a given subdivision
of the interval [a, b] such that hi = xi+1 − xi = h = b−a
and let the assumptions
n
of Theorem 1.4 hold. Then we have
Z
n−1 b
xi + xi+1
hX
f (xi ) + 2f
(3.1) f (t) dt −
+ f (xi+1 ) a
4 i=0
2
3
Zheng Liu
3
(b − a) 2
(b − a) 2
δn (f ) ≤ √
λn (f ).
≤ √
4 3n
4 3n
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where
[f (b) − f (a)]2
(3.2) δn (f ) = kf 0 k22 −
b−a
1
"
# 2  2
n−1
n−1 X
X
1
xi + xi+1
−
f (x0 ) + f (xn ) + 2
f (xi ) − 2
f

b−a
2
i=1
i=0
and
(3.3)
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1
[f (b) − f (a)]2 2
0 2
λn (f ) = kf k2 −
.
b−a
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Proof. From (1.10) and (1.11) in Theorem 1.4 we obtain
Z xi+1
h
x
+
x
i
i+1
(3.4) f (xi ) + 2f
+ f (xi+1 ) −
f (t) dt
4
2
xi
Z xi+i
3
h2
1
≤ √
(f 0 (t))2 dt − [f (xi+1 ) − f (xi )]2
h
4 3
xi
2 ) 12
xi + xi+1
1
.
−
f (xi ) − 2f
+ f (xi+1 )
h
2
By summing (3.4) over i from 0 to n − 1 and using the generalized triangle
inequality, we get
Z
n−1 b
hX
xi + xi+1
(3.5) f (xi ) + 2f
+ f (xi+1 ) f (t) dt −
a
4 i=0
2
3
n−1 Z xi+i
h2 X
1
≤ √
(f 0 (t))2 dt − [f (xi+1 ) − f (xi )]2
h
4 3 i=0
xi
2 ) 12
1
xi + xi+1
f (xi ) − 2f
+ f (xi+1 )
−
.
h
2
A Unified Treatment of Some
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Zheng Liu
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By using the Cauchy inequality twice, we can easily obtain
(3.6)
n−1 Z
X
i=0
xi+1
xi
1
(f 0 (t))2 dt − [f (xi+1 ) − f (xi )]2
h
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2 ) 12
1
xi + xi+1
− f (xi ) − 2f
+ f (xi+1 )
h
2
(
n−1
√
n X
0 2
≤ n kf k2 −
[f (xi+1 ) − f (xi )]2
b − a i=0
2 ) 12
n−1 xi + xi+1
n X
−
f (xi ) − 2f
+ f (xi+1 )
b − a i=0
2
√
[f (b) − f (a)]2
≤ n kf 0 k22 −
b−a
1
"
# 2  2
n−1
n−1 X
X
xi + xi+1
1
−
f (x0 ) + f (xn ) + 2
f (xi ) − 2
f
.

b−a
2
i=1
i=0
Consequently, the inequality (3.1) with (3.2) and (3.3) follow from (3.5) and
(3.6).
Remark 3. It should be noticed that Theorem 3.1 seems to be a revision and an
improvement of the corresponding result in [2, Theorem 6.1].
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
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References
[1] N. UJEVIĆ, A generalization of the pre-Grüss inequality and applications to some qradrature formulae, J. Inequal. Pure Appl. Math., 3(1)
(2002), Art. 13. [ONLINE: http://jipam.vu.edu.au/article.
php?sid=165].
[2] N. UJEVIĆ, Sharp error bounds for some quadrature formulae and applications, J. Inequal. Pure Appl. Math., 7(1) (2006), Art. 8. [ONLINE:
http://jipam.vu.edu.au/article.php?sid=622].
A Unified Treatment of Some
Sharp Inequalities
Zheng Liu
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