In this paper, a generalized perturbed midpoint rule is established. Various error bounds for this generalization are also obtained.
AMS Mathematics Subject Classification : 26D15, 26D30.
1. Introduction
In recent years a number of authors have considered error inequalities for some known and some new quadrature rules. Sometimes they have considered generalizations of these rules. For example, the wellknown trapezoid and midpoint quadrature rules are considered in (
[1
,
2
,
9
,
10]
). In
[2]
, we can find
where
denotes the integer part of
and
For
n
= 1, we get the midpoint rule
In
[11]
, a generalized trapezoid rule is derived by Ujevi
ć
as follows:
where
In
[4]
and
[8]
, the following unified treatment for generalizations of the midpoint, trapezoid, averaged midpointtrapezoid and Simpson type inequalities is obtained by Liu and Liu, respectively,
where
θ
∈ [0, 1] and
In
[5]
, Liu established the following generalized perturbed trapezoid rule.
where
K_{n}
(
x
) is the kernel given by
In
[7]
and
[12]
, the following generalization of the perturbed midpointtrapezoid rule is established by Liu and Ujevi
ć
et al., respectively.
Theorem 1.1.
Let f
: [
a, b
] → ℝ
be a function such that f
^{(n−1)}
is absolutely continuous on
[
a, b
].
Then we have
where
denotes the integer part of
and R
(
f
) = (−1)
^{n}
Some sharp perturbed midpoint inequalities are proved by Liu in
[6]
based on the following identity:
where
and
Theorem 1.2
(
[6]
).
Let f
: [
a, b
] → ℝ
be a twice differentiable mapping such that f′′ is integrable with
Γ
_{2}
= sup
_{x∈(a,b)}
f′′
(
x
)
and
γ
_{2}
= inf
_{x∈(a,b)}
f′′
(
x
).
Then we have
Theorem 1.3
(
[6]
).
Let f
: [
a, b
] → ℝ
be a thirdorder differentiable mapping such that f′′′ is integrable with
Γ
_{3}
= sup
_{x∈(a,b)}
f′′′
(
x
)
and
γ
_{3}
= inf
_{x∈(a,b)}
f′′′
(
x
).
Then we have
The purpose of this paper is to extend (2) to a more general version, that is, a generalized perturbed midpoint rule is established. Various error bounds for the generalizations are also given.
2. For differentiable mappings with bounded derivatives
Theorem 2.1.
Let f
: [
a, b
] → ℝ
be a mapping such that the derivative f
^{(n−1)}
(
n
≥ 2)
is absolutely continuous on
[
a, b
]
and M_{n}
= sup
_{x∈(a,b)}

f
^{(n)}
(
x
) < ∞.
Then we have
where
denotes the integer part of
.
Proof
. It is not difficult to find the identity
where
S_{n}
(
x
) is the kernel given by
Using the above identity, we get
Now, we put
It is clear that
P_{n}
(
x
) and
Q_{n}
(
x
) are symmetric with respect to the line
for n even and symmetric with respect to the point
for n odd. Therefore,
By substitution
we find that
is always negative on [0, 1] for
n
≥ 3. Thus
for
n
≥ 3, and
Hence,
Consequently, inequalities (9) follow from (11) and (12).
Remark
. Applying (10) for
n
= 2, 3 respectively, we get the identity (2).
For convenience in further discussions, we collect some technical results which are not difficult to obtain by elementary calculus as:
Before we end this section, we introduce the notations
3. For functions whose (n − 1)th derivatives are Lipschitzian type
Recall that a function
f
: [
a, b
] →
R
is said to be
L
Lipschitzian on [
a, b
] if
for all
x, y
∈ [
a, b
],where
L
> 0 is given, and, it is said to be (
l,L
) Lipschitzian on [
a, b
] if
for all
a
≤
x
≤
y
≤
b
where
l,L
∈
R
with
l
<
L
.
From
[3]
, we get that if
h, g
: [
a, b
] → ℝ are such that
h
is Riemannintegral on [
a, b
] and
g
is
L
Lipschitzian on [
a, b
], then
exists and
Theorem 3.1.
Let f
: [
a, b
] → ℝ
be a mapping such that derivative f
^{(n−1)}
(
n
≥ 2)
is
(
l,L
)
Lipschitzian on
[
a, b
].
Then we have
Proof
. By (10) and (13) we get
Then notice that
Lipschitzian on [
a, b
] and by using (16), we have
Hence, the inequality (17) follows from (16) and (12).
Corollary 3.2.
Let f
: [
a, b
] → ℝ
be a mapping such that derivative f
^{(n−1)}
(
n
≥ 2)
is
L

Lipschitzian on
[
a, b
].
Then we have
4. Bounds in terms of some Lebesgue norms
Theorem 4.1.
Let f
: [
a, b
] → ℝ
be a mapping such that the
(
n
1)
th derivative f
^{(n−1)}
(
n
≥ 2)
is
absolutely continuous on
[
a, b
].
If f
^{(n)}
∈
L
_{∞}
[
a, b
],
then we have
where
∥
f
^{(n)}
∥
_{∞}
:=
ess
sup
_{x∈[a,b]}
f
^{(n)}
(
x
)
is the usual Lebesgue norm on L
_{∞}
[
a, b
].
Proof
. We can obtain the result by taking
L
= ∥
f
^{(n)}
∥
_{∞}
in Corollary 3.2.
Theorem 4.2.
Let f
: [
a, b
] → ℝ
be a mapping such that the
(
n
1)
th derivative f
^{(n−1)}
(
n
≥ 2)
is
absolutely continuous on
[
a, b
].
If f
^{(n)}
∈
L
_{1}
[
a, b
],
then we have
where
is the usual Lebesgue norm on L
_{1}
[
a, b
].
Proof
. By using the identity (10) we get
Then the conclusion follows from (15).
Theorem 4.3.
Let f
: [
a, b
] → ℝ
be a mapping such that the
(
n
1)
th derivative f
^{(n−1)}
(
n
≥ 2)
is
absolutely continuous on
[
a, b
].
If f
^{(n)}
∈
L
_{2}
[
a, b
],
then we have
where
is the usual Lebesgue norm on L
_{2}
[
a, b
].
Proof
. By using the identity (10) we get
Then the conclusion follows from (14).
5. Non symmetric bounds
Theorem 5.1.
Let f
: [
a, b
] → ℝ
be a mapping such that the
(
n
1)
th derivative f
^{(n−1)}
(
n
≥ 2)
is
absolutely continuous with
γ_{n}
≤
f
^{(n)}
(
x
) ≤ Γ
_{n}
a.e. on
[
a, b
],
where
γ_{n}
, Γ
_{n}
∈ ℝ
are constants, then we have
Proof
. By (10) and (13) we get
then notice that
a.e. on [
a, b
], we have
We complete the proof from (12).
Remark
. Applying Theorem 5.1 for
n
= 2, 3, we get (3), (6), respectively.
Theorem 5.2.
Let f
: [
a, b
] → ℝ
be a mapping such that the
(
n
1)
th derivative f
^{(n−1)}
(
n
≥ 2)
is
absolutely continuous with
γ_{n}
≤
f
^{(n)}
(
x
) ≤ Γ
_{n}
a.e. on
[
a, b
],
where
γ_{n}
∈ ℝ
is a constant, then we have
where
Proof
. By (10) and (13) we get
then notice that
f
^{(n)}
(
x
) −
γ_{n}
≥ 0 a.e. on [
a, b
], we have
From (15), we get the desired result.
Remark.
Applying Theorem 5.2 for
n
= 2, 3, we get (4), (7), respectively.
Theorem 5.3.
Let f
: [
a, b
] → ℝ
be a mapping such that the
(
n
−1)
th derivative f
^{(n−1)}
(
n
≥ 2)
is absolutely continuous with f
^{(n)}
(
x
) ≤ Γ
_{n}
a.e. on
[
a, b
],
where
Γ
_{n}
∈ ℝ
is a constant, then we have
where D_{n} is defined in Theorem 5.2.
Proof
. The proof of inequalities (18) is similar to the proof of Theorem 5.2 and so is omitted.
Remark
. Applying Theorem 5.3 for
n
= 2, 3, we get (5), (8), respectively.
6. Another sharp bound
In this section, we derive two sharp error inequalities when n is an odd and an even integer, respectively.
Theorem 6.1.
Let f
: [
a, b
] → ℝ
be a mapping such that the
(
n
−1)
th derivative f
^{(n−1)}
(
n
≥ 2)
is absolutely continuous on
[
a, b
].
If f
^{(n)}
∈
L
_{2}
[
a, b
]
and n is an odd integer. Then we have
where σ(·) is defined by
Inequality (19) is the best possible in the sense that the constant
can not be replaced by a smaller one.
Proof
. By using the identity (10) and (13) we get
To prove the sharpness of (19), we suppose that (19) holds with a constant
C
> 0 as
We may find a function
f
: [
a, b
] → ℝ such that the (
n
− 1)th derivative
f
^{(n−1)}
(
n
≥ 2) is absolutely continuous on [
a, b
] as
It follows that
Then we can find that the lefthand side of inequality (20) becomes
and the righthand side of inequality (20) becomes
From (20), (21) and (22), we get
which proving that the constant
is the best possible in (19).
Theorem 6.2.
Let f
: [
a, b
] → ℝ
be a mapping such that the
(
n
−1)
th derivative f
^{(n−1)}
(
n
≥ 2)
is absolutely continuous on
[
a, b
].
If f
^{(n)}
∈
L
_{2}
[
a, b
]
and n is an even integer. Then we have
where σ
(·)
is defined in Theorem 6.1. Inequality (23) is the best possible in the sense that the constant
can not be replaced by a smaller one.
Proof
. By using the identity (10) and (13) we get
We now suppose that (23) holds with a constant
C
> 0 as
We may find a function
f
: [
a, b
] → ℝ such that the (
n
− 1)th derivative
f
^{(n−1)}
(
n
≥ 2) is absolutely continuous on [
a, b
] as
It follows that
Then we can find that the lefthand side of inequality (24) becomes
and the righthand side of inequality (24) becomes
It follows from (24), (25) and (26) that
proving that the constant
is the best possible in (23).
Acknowledgements
This work is supported by Youth Project of Chongqing Three Gorges University of China (No.13QN11).
BIO
Feixiang Chen received his MS degree from Xi'an Jiaotong University. Since 2009 he has been teaching at Chongqing Three Gorges University. His research interests include integral inequalities on several kinds of convex functions and applications.
School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, 404000, P.R. China.
email:cfx2002@126.com
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