SOME NEW INEQUALITIES OF PERTURBED MIDPOINT RULE

Abstract

In this paper, a generalized perturbed midpoint rule is established. Various error bounds for this generalization are also obtained.

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 well-known 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 Ujević as follows:

where

In [4] and [8], the following unified treatment for generalizations of the midpoint, trapezoid, averaged midpoint-trapezoid 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 Kn(x) is the kernel given by

In [7] and [12], the following generalization of the perturbed midpoint-trapezoid rule is established by Liu and Ujević 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 Γ2 = supx∈(a,b) f′′(x) and γ2 = infx∈(a,b) f′′(x). Then we have

Theorem 1.3 ([6]). Let f : [a, b] → ℝ be a third-order differentiable mapping such that f′′′ is integrable with Γ3 = supx∈(a,b) f′′′(x) and γ3 = infx∈(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 Mn = supx∈(a,b) |f(n)(x)| < ∞. Then we have

where denotes the integer part of .

Proof. It is not difficult to find the identity

where Sn(x) is the kernel given by

Using the above identity, we get

Now, we put

It is clear that Pn(x) and Qn(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 Riemann-integral 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 supx∈[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) ∈ L1[a, b], then we have

where is the usual Lebesgue norm on L1[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) ∈ L2[a, b], then we have

where is the usual Lebesgue norm on L2[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 γn ≤ f(n)(x) ≤ Γn a.e. on [a, b], where γn, Γn ∈ 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 γn ≤ f(n)(x) ≤ Γn a.e. on [a, b], where γn ∈ ℝ is a constant, then we have

where

Proof. By (10) and (13) we get

then notice that f(n)(x) − γn ≥ 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) ≤ Γn a.e. on [a, b], where Γn ∈ ℝ is a constant, then we have

where Dn 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) ∈ L2[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 left-hand side of inequality (20) becomes

and the right-hand 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) ∈ L2[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 left-hand side of inequality (24) becomes

and the right-hand side of inequality (24) becomes

It follows from (24), (25) and (26) that

proving that the constant is the best possible in (23).