Journal of Inequalities in Pure and
Applied Mathematics
SOME INEQUALITIES OF PERTURBED TRAPEZOID TYPE
ZHENG LIU
Institute of Applied Mathematics
Faculty of Science
Anshan University of Science and Technology
Anshan 114044, Liaoning, China.
volume 7, issue 2, article 47,
2006.
Received 21 June, 2005;
accepted 16 March, 2006.
Communicated by: J.E. Pečarić
EMail: lewzheng@163.net
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
186-05
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Abstract
A new generalized perturbed trapezoid type inequality is established by Peano
kernel approach. Some related results are also given.
2000 Mathematics Subject Classification: 26D15.
Key words: Perturbed trapezoid inequality; n-times continuously differentiable mapping; Absolutely continuous.
Contents
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
For Differentiable Mappings With Bounded Derivatives . . . . 5
3
Bounds In Terms of Some Lebesgue Norms . . . . . . . . . . . . . . 10
4
Non-Symmetric Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
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1.
Introduction
In recent years, some authors have considered the perturbed trapezoid inequality
Z b
2
(b
−
a)
b
−
a
0
0
f (x) dx −
[f (a) + f (b)] +
[f (b) − f (a)]
2
12
a
≤ C(Γ2 − γ2 )(b − a)3 ,
where f : [a, b] → R is a twice differentiable mapping on (a, b) with γ2 =
inf x∈[a,b] f 00 (x) > −∞ and Γ2 = supx∈[a,b] f 00 (x) < +∞ while C is a constant.
√
(e.g. see [1] – [8]) It seems that the best result C = 1083 was separately and
independently discovered by the authors of [5] and [8]. The perturbed trapezoid
inequality has been established as
(1.1)
Z b
(b − a)2 0
b−a
0
[f
(a)
+
f
(b)]
+
[f
(b)
−
f
(a)]
f
(x)
dx
−
2
12
a
√
3
≤
(Γ2 − γ2 )(b − a)3 .
108
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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Moreover, we can also find in [5] the following two perturbed trapezoid inequalities as
Z b
b−a
(b − a)2 0
0
(1.2) f (x) dx −
[f (a) + f (b)] +
[f (b) − f (a)]
2
12
a
1
≤
(Γ3 − γ3 )(b − a)4 ,
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where f : [a, b] → R is a third-order differentiable mapping on (a, b) with
γ3 = inf x∈[a,b] f 000 (x) > −∞ and Γ3 = supx∈[a,b] f 000 (x) < +∞, and
(1.3)
Z b
2
b
−
a
(b
−
a)
0
0
[f
(a)
+
f
(b)]
+
[f
(b)
−
f
(a)]
f
(x)
dx
−
2
12
a
1
≤
M4 (b − a)5 ,
720
where f : [a, b] → R is a fourth-order differentiable mapping on (a, b) with
M4 = sup |f (4) (x)| < +∞.
Some Inequalities of Perturbed
Trapezoid Type
x∈[a,b]
The purpose of this paper is to extend these above results to a more general
version by choosing appropriate harmonic polynomials such as the Peano kernel. A new generalized perturbed trapezoid type inequality is established and
some related results are also given.
Zheng Liu
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2.
For Differentiable Mappings With Bounded
Derivatives
Theorem 2.1. Let f : [a, b] → R be an n-times continuously differentiable
mapping, n ≥ 2 and such that Mn := supx∈[a,b] |f (n) (x)| < ∞. Then
Z b
(b − a)2 0
b−a
[f (a) + f (b)] +
[f (b) − f 0 (a)]
(2.1) f (x) dx −
2
12
a
]
[ n−1
2
2k+1
X
a + b k(k − 1)(b − a)
−
f (2k)
2k−2
3(2k + 1)!2
2
k=2
 √
3

if n = 2;
 3(b−a)
54
≤ Mn ×

 n(n−2)(b−a)nn+1 if n ≥ 3,
3(n+1)!2
where [ n−1
] denotes the integer part of
2
n−1
.
2
Proof. It is not difficult to find the identity
Z b
n
(2.2) (−1)
Tn (x)f (n) (x) dx
a
Z b
b−a
(b − a)2 0
=
f (x) dx −
[f (a) + f (b)] +
[f (b) − f 0 (a)]
2
12
a
[ n−1
]
2
X
k(k − 1)(b − a)2k+1 (2k) a + b
−
f
,
2k−2
3(2k
+
1)!2
2
k=2
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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where Tn (x) is the kernel given by
(2.3) Tn (x) =


(x−a)n
n!
−
(b−a)(x−a)n−1
2(n−1)!
+
(b−a)2 (x−a)n−2
12(n−2)!
if x ∈ a, a+b
,
2

(x−b)n
n!
+
(b−a)(x−b)n−1
2(n−1)!
+
(b−a)2 (x−b)n−2
12(n−2)!
if x ∈
a+b
,b
2
.
Using the identity (2.2), we get
(2.4)
Z b
(b − a)2 0
b−a
[f
(a)
+
f
(b)]
+
[f (b) − f 0 (a)]
f
(x)
dx
−
2
12
a
]
[ n−1
2
2k+1
X k(k − 1)(b − a)
a + b (2k)
−
f
3(2k + 1)!22k−2
2
k=2
Z b
Z b
(n)
=
Tn (x)f (x) dx ≤ Mn
|Tn (x)| dx.
a
a
For brevity, we put
(x − a)n (b − a)(x − a)n−1 (b − a)2 (x − a)n−2
−
+
n!
2(n − 1)!
12(n − 2)!
n−2
n(b − a)(x − a) n(n − 1)(b − a)2
(x − a)
2
=
(x − a) −
+
,
n!
2
12
a+b
x ∈ a,
2
Pn (x) :=
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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and
(x − b)n (b − a)(x − b)n−1 (b − a)2 (x − b)n−2
+
+
Qn (x) :=
n!
2(n − 1)!
12(n − 2)!
n−2
(x − b)
n(b − a)(x − b) n(n − 1)(b − a)2
2
=
(x − b) +
+
,
n!
2
12
a+b
x∈
,b .
2
It is clear that Pn (x) and Qn (x) are symmetric with respect to the line x =
a+b
for n even, and symmetric with respect to the point ( a+b
, 0) for n odd.
2
2
Therefore,
Z
b
Z
|Tn (x)| dx = 2
a
a+b
2
|Pn (x)| dx
a
Z (b − a)n+1 1 n−2 2
n(n − 1) =
t − nt +
t
dt
n!2n
3
0
by substitution x = a +
b−a
t,
2
and it is easy to find that
n(n − 1)
n−2
2
rn (t) := t
t − nt +
3
is always nonnegative on [0, 1] for n ≥ 3. Thus we have
Z 1
Z 1
n(n − 1)
n(n − 2)
n−2
2
|rn (t)| dt =
t
t − nt +
dt =
3
3(n + 1)
0
0
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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for n ≥ 3, and
Z 1
Z 1
2
2
t − 2t + dt
|r2 (t)| dt =
3
0
0
Z 1
Z t0
2
2
2
2
=
t − 2t +
dt −
t − 2t +
dt,
3
3
0
t0
√
where t0 = 1 −
3
3
Z
is the unique zero of r2 (t) in (0, 1). Hence,
b
|Tn (x)| dx =
(2.5)
a



√
3(b−a)3
,
54
n = 2,
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
n(n−2)(b−a)n+1
,
3(n+1)!2n
n ≥ 3.
Consequently, the inequality (2.1) follows from (2.4) and (2.5).
Remark 1. If in the inequality (2.1) we choose n = 2, 3, 4, then we get
Z b
√
(b − a)2 0
3
b−a
0
[f (a) + f (b)] +
[f (b) − f (a)] ≤
M2 (b − a)3 ,
f (x) dx −
2
12
54
Za b
2
b
−
a
(b
−
a)
0
0
≤ 1 M3 (b − a)4
f
(x)
dx
−
[f
(a)
+
f
(b)]
+
[f
(b)
−
f
(a)]
192
2
12
a
and the inequality (1.3), respectively.
For convenience in further discussions, we will now collect some technical
results related to (2.3) which are not difficult to obtain by elementary calculus
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as:
Z
b
Tn (x) dx =
(2.6)
a
(2.7)
max |Tn (x)| =
x∈[a,b]

 0,












n odd,
n(n−2)(b−a)n+1
,
3(n+1)!2n
(b−a)2
,
12
√
n even.
n = 2,
3(b−a)3
,
216
Some Inequalities of Perturbed
Trapezoid Type
n = 3,
(n−1)(n−3)(b−a)n
,
3(n!)2n
Zheng Liu
n ≥ 4.
Title Page
(2.8)
Z b
1
max T2m (x) −
T2m (x) dx
x∈[a,b]
b−a a
 (b−a)4
 720 ,
=
 (8m3 −16m2 +2m+3)(b−a)2m ,
3(2m+1)!22m
Contents
m = 2,
m ≥ 3.
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3.
Bounds In Terms of Some Lebesgue Norms
Theorem 3.1. Let f : [a, b] → R be a mapping such that the derivative f (n−1)
(n ≥ 2) is absolutely continuous on [a, b]. If f (n) ∈ L∞ [a, b], then we have
(3.1)
Z b
b−a
(b − a)2 0
f
(x)
dx
−
[f
(a)
+
f
(b)]
+
[f (b) − f 0 (a)]
2
12
a
[ n−1
]
2
2k+1
X
k(k − 1)(b − a)
a + b −
f (2k)
2k−2
3(2k + 1)!2
2
k=2
 √3(b−a)3

,
n = 2,
54
≤ kf (n) k∞ ×
 n(n−2)(b−a)n+1
, n ≥ 3,
3(n+1)!2n
where [ n−1
] denotes the integer part of n−1
and kf (n) k∞ := ess supx∈[a,b] |f (n) (x)|
2
2
is the usual Lebesgue norm on L∞ [a, b].
The proof of inequality (3.1) is similar to the proof of inequality (2.1) and so
is omitted.
Theorem 3.2. Let f : [a, b] → R be a mapping such that the derivative
f (n−1) (n ≥ 2) is absolutely continuous on [a, b]. If f (n) ∈ L1 [a, b], then we
have
Z b
(b − a)2 0
b−a
[f (a) + f (b)] +
[f (b) − f 0 (a)]
(3.2) f (x) dx −
2
12
a
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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X k(k − 1)(b − a)2k+1
a + b (2k)
−
f
3(2k + 1)!22k−2
2
k=2
 (b−a)2
,


12


 √
3(b−a)3
≤ kf (n) k1 ×
,
216




 (n−1)(n−3)(b−a)n
[ n−1
]
2
3(n!)2n
where kf (n) k1 :=
Rb
a
n = 2,
n = 3,
,
n ≥ 4,
Some Inequalities of Perturbed
Trapezoid Type
|f (x)| dx is the usual Lebesgue norm on L1 [a, b].
Zheng Liu
Proof. By using the identity (2.2), we get
Z b
(b − a)2 0
b−a
[f
(a)
+
f
(b)]
+
[f (b) − f 0 (a)]
f
(x)
dx
−
2
12
a
[ n−1
]
2
2k+1
X
k(k − 1)(b − a)
a + b −
f (2k)
2k−2
3(2k + 1)!2
2
k=2
Z b
Z b
= Tn (x)f (n) (x) dx ≤ max |Tn (x)|
|f (n) (x)| dx.
a
Then the inequality (3.2) follows from (2.7).
x∈[a,b]
a
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4.
Non-Symmetric Bounds
Theorem 4.1. Let f : [a, b] → R be a mapping such that the derivative
f (n) (n ≥ 2) is integrable with γn = inf x∈[a,b] f (n) (x) > −∞ and Γn =
supx∈[a,b] f (n) (x) < +∞. Then we have
(4.1)
(4.2)
Z b
(b − a)2 0
b−a
f
(x)
dx
−
[f
(a)
+
f
(b)]
+
[f (b) − f 0 (a)]
2
12
a
]
[ n−1
2
2k+1
X
k(k − 1)(b − a)
a + b (2k)
f
−
3(2k + 1)!22k−2
2
k=2
 √
3(b−a)3

,
n = 2,

54
Γn − γn
≤
×

2
 n(n−2)(b−a)nn+1 , n ≥ 3 and odd,
3(n+1)!2
Z b
b−a
(b − a)2 0
f
(x)
dx
−
[f
(a)
+
f
(b)]
+
[f (b) − f 0 (a)]
2
12
a
[ n−1
]
2
2k+1
X k(k − 1)(b − a)
a + b (2k)
−
f
2k−2
3(2k
+
1)!2
2
k=2
≤ [f (n−1) (b) − f (n−1) (a) − γn (b − a)]
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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×









(4.3)
(b−a)2
,
12
√
3(b−a)3
,
216
(n−1)(n−3)(b−a)n
,
3(n!)2n
n = 2,
n = 3,
n ≥ 5 and odd,
Z b
(b − a)2 0
b−a
f (x) dx −
[f (a) + f (b)] +
[f (b) − f 0 (a)]
2
12
a
]
[ n−1
2
2k+1
X
k(k − 1)(b − a)
a + b (2k)
f
−
3(2k + 1)!22k−2
2
k=2
≤ [Γn (b − a) − f (n−1) (b) + f (n−1) (a)]
 (b−a)2
,
n = 2,


12

 √
3(b−a)3
×
,
n = 3,
216



n
 (n−1)(n−3)(b−a)
, n ≥ 5 and odd,
3(n!)2n
(4.4)
Z b
(b − a)2 0
b−a
[f
(a)
+
f
(b)]
+
[f (b) − f 0 (a)]
f
(x)
dx
−
2
12
a
[ n−1
]
2
X
k(k − 1)(b − a)2k+1 (2k) a + b
−
f
2k−2
3(2k
+
1)!2
2
k=2
m(m − 1)(b − a)2m (2m−1)
(2m−1)
−
[f
(b)
−
f
(a)]
3(2m + 1)!22m−2
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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≤ [f (2m−1) (b) − f (2m−1) (a) − γ2m (b − a)]
 (b−a)4
 720 ,
×
 (8m3 −16m2 +2m+3)(b−a)2m
3(2m+1)!22m
m = 2,
,
m ≥ 3,
Z b
(b − a)2 0
b−a
(4.5) f (x) dx −
[f (a) + f (b)] +
[f (b) − f 0 (a)]
2
12
a
n−1
[ 2 ]
X k(k − 1)(b − a)2k+1
a+b
(2k)
f
−
3(2k + 1)!22k−2
2
k=2
m(m − 1)(b − a)2m (2m−1)
(2m−1)
−
[f
(b)
−
f
(a)]
3(2m + 1)!22m−2
 (b−a)4
m = 2,
 720 ,
≤ [Γ2m (b−a)−f (2m−1) (b)+f (2m−1) (a)]
 (8m3 −16m2 +2m+3)(b−a)2m
, m ≥ 3.
3(2m+1)!22m
Proof. For n odd and n = 2, by (2.2) and (2.6) we get
Z b
n
(−1)
Tn (x)[f (n) (x) − C] dx
a
Z b
(b − a)2 0
b−a
=
[f (a) + f (b)] +
[f (b) − f 0 (a)]
f (x) dx −
2
12
a
[ n−1
]
2
X
k(k − 1)(b − a)2k+1 (2k) a + b
−
f
.
2k−2
3(2k
+
1)!2
2
k=2
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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where C ∈ R is a constant.
n
If we choose C = γn +Γ
, then we have
2
Z b
b−a
(b − a)2 0
f (x) dx −
[f (a) + f (b)] +
[f (b) − f 0 (a)]
2
12
a
n−1
[ 2 ]
X k(k − 1)(b − a)2k+1
a + b (2k)
−
f
3(2k + 1)!22k−2
2
k=2
Z
Γn − γn b
≤
|Tn (x)| dx.
2
a
and hence the inequality (4.1) follows from (2.5).
If we choose C = γn , then we have
Zheng Liu
Title Page
Z b
(b − a)2 0
b−a
[f
(a)
+
f
(b)]
+
[f (b) − f 0 (a)]
f
(x)
dx
−
2
12
a
[ n−1
]
2
2k+1
X
k(k − 1)(b − a)
a + b (2k)
f
−
3(2k + 1)!22k−2
2
k=2
Z b
≤ max |Tn (x)|
|f (n) (x) − γn | dx,
x∈[a,b]
Some Inequalities of Perturbed
Trapezoid Type
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a
and hence the inequality (4.2) follows from (2.7).
Similarly we can prove that the inequality (4.3) holds.
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By (2.2) and (2.6) we can also get
Z b
b−a
(b − a)2 0
f (x) dx −
[f (a) + f (b)] +
[f (b) − f 0 (a)]
2
12
a
[ n−1
]
2
X
k(k − 1)(b − a)2k+1 (2k) a + b
−
f
3(2k + 1)!22k−2
2
k=2
m(m − 1)(b − a)2m (2m−1)
(2m−1)
−
[f
(b)
−
f
(a)]
3(2m + 1)!22m−2
Z b Z b
1
2m
= T2m (x) −
T2m (x) dx [f (x) − C] dx ,
b−a a
a
where C ∈ R is a constant.
If we choose C = γ2m , then we have
Z b
(b − a)2 0
b−a
f
(x)
dx
−
[f
(a)
+
f
(b)]
+
[f (b) − f 0 (a)]
2
12
a
[ n−1
]
2
X
k(k − 1)(b − a)2k+1 (2k) a + b
−
f
2k−2
3(2k
+
1)!2
2
k=2
m(m − 1)(b − a)2m (2m−1)
(2m−1)
−
[f
(b)
−
f
(a)]
3(2m + 1)!22m−2
Z b
Z b
1
≤ max T2m (x) −
T2m (x) dx
|f (2m) (x) − γ2m | dx
x∈[a,b]
b−a a
a
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and hence the inequality (4.4) follows from (2.8).
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Similarly we can prove that the inequality (4.5) holds.
Remark 2. It is not difficult to find that the inequality (4.1) is sharp in the sense
that we can choose f to attain
R xtheR yequality in (4.1). Indeed, for n = 2, we
construct the function f (x) = a a j(z) dz dy, where
√
√

(3+ 3)a+(3− 3)b

,
Γ
,
a
≤
x
<
2

6



√
√
√
√
3)b
(3− 3)a+(3+ 3)b
j(x) =
γ2 , (3+ 3)a+(3−
≤
x
<
,
6
6



√
√


3)b
Γ2 , (3− 3)a+(3+
≤ x ≤ b,
6
and for n ≥ 3 and odd, we construct the function
Z x Z yn Z y2
f (x) =
···
j(y1 ) dy1 · · · dyn−1 dyn ,
a
a
where
j(x) =
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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a

 Γn ,
a≤x<
 γ ,
n
a+b
2
a+b
,
2
≤ x ≤ b.
Remark 3. If in the inequality (4.1) we choose n = 2, 3, then we recapture the
inequalities (1.1) and (1.2), respectively.
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References
[1] P. CERONE, On perturbed trapezoidal and midpoint rules, Korean J. Comput. Appl. Math., 2 (2002), 423–435.
[2] P. CERONE AND S.S. DRAGOMIR, Trapezoidal type rules from an inequalities point of view, Handbook of Analytic-Computational Methods in
Applied Mathematics, CRC Press N.Y.(2000), 65–134.
[3] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, An inequality of
Ostrowski-Grüss type for twice differentiable mappings and applications in
numerical integration, Kyungpook Math. J., 39 (1999), 331–341.
[4] X.L. CHENG, Improvement of some Ostrowski-Grüss type inequalities,
Comput. Math. Appl., 42 (2001), 109–114.
[5] X.L. CHENG AND J. SUN, A note on the perturbed trapezoid inequality, J.
Inequal. in Pure and Appl. Math., 3(2) (2002), Art. 29. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=181].
[6] S.S. DRAGOMIR, P. CERONE AND A. SOFO, Some remarks on the trapezoid rule in numerical integration, Indian J. of Pure and Appl. Math., 31(5)
(2000), 489–501.
[7] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, Improvement and further generalization of inequalities of Ostrowski-Grüss type, Computer Math. Appl.,
39 (2000), 161–175.
[8] N. UJEVIĆ, On perturbed mid-point and trapezoid inequalities and applications, Kyungpook Math. J., 43 (2003), 327–334.
Some Inequalities of Perturbed
Trapezoid Type
Zheng Liu
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