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SOME NEW INEQUALITIES OF PERTURBED MIDPOINT RULE
SOME NEW INEQUALITIES OF PERTURBED MIDPOINT RULE
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 635-647
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : December 10, 2013
  • Accepted : July 14, 2014
  • Published : September 30, 2014
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FEIXIANG CHEN

Abstract
In this paper, a generalized perturbed midpoint rule is estab-lished. Various error bounds for this generalization are also obtained. AMS Mathematics Subject Classification : 26D15, 26D30.
Keywords
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
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where
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denotes the integer part of
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and
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For n = 1, we get the midpoint rule
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In [11] , a generalized trapezoid rule is derived by Ujevi ć as follows:
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where
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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,
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where θ ∈ [0, 1] and
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In [5] , Liu established the following generalized perturbed trapezoid rule.
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where Kn ( x ) is the kernel given by
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In [7] and [12] , the following generalization of the perturbed midpoint-trapezoid 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
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where
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denotes the integer part of
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and R ( f ) = (−1) n
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PPT Slide
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Some sharp perturbed midpoint inequalities are proved by Liu in [6] based on the following identity:
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where
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and
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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
PPT Slide
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PPT Slide
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PPT Slide
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Theorem 1.3 ( [6] ). Let f : [ a, b ] → ℝ be a third-order 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
PPT Slide
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PPT Slide
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PPT Slide
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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 = sup x∈(a,b) | f (n) ( x )| < ∞. Then we have
PPT Slide
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where
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denotes the integer part of
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.
Proof . It is not difficult to find the identity
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where Sn ( x ) is the kernel given by
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Using the above identity, we get
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Now, we put
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It is clear that Pn ( x ) and Qn ( x ) are symmetric with respect to the line
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for n even and symmetric with respect to the point
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for n odd. Therefore,
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By substitution
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we find that
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is always negative on [0, 1] for n ≥ 3. Thus
PPT Slide
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for n ≥ 3, and
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Hence,
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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:
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Before we end this section, we introduce the notations
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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
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for all x, y ∈ [ a, b ],where L > 0 is given, and, it is said to be ( l,L ) -Lipschitzian on [ a, b ] if
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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
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exists and
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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
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Proof . By (10) and (13) we get
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Then notice that
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Lipschitzian on [ a, b ] and by using (16), we have
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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
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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
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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
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where
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is the usual Lebesgue norm on L 1 [ a, b ].
Proof . By using the identity (10) we get
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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
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where
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is the usual Lebesgue norm on L 2 [ a, b ].
Proof . By using the identity (10) we get
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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 ) ≤ Γ n a.e. on [ a, b ], where γn , Γ n ∈ ℝ are constants, then we have
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Proof . By (10) and (13) we get
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then notice that
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a.e. on [ a, b ], we have
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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 ) ≤ Γ n a.e. on [ a, b ], where γn ∈ ℝ is a constant, then we have
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where
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Proof . By (10) and (13) we get
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then notice that f (n) ( x ) − γn ≥ 0 a.e. on [ a, b ], we have
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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
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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) L 2 [ a, b ] and n is an odd integer. Then we have
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where σ(·) is defined by
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Inequality (19) is the best possible in the sense that the constant
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can not be replaced by a smaller one.
Proof . By using the identity (10) and (13) we get
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To prove the sharpness of (19), we suppose that (19) holds with a constant C > 0 as
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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
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It follows that
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Then we can find that the left-hand side of inequality (20) becomes
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and the right-hand side of inequality (20) becomes
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From (20), (21) and (22), we get
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which proving that the constant
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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
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where σ (·) is defined in Theorem 6.1. Inequality (23) is the best possible in the sense that the constant
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can not be replaced by a smaller one.
Proof . By using the identity (10) and (13) we get
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We now suppose that (23) holds with a constant C > 0 as
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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
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It follows that
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Then we can find that the left-hand side of inequality (24) becomes
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and the right-hand side of inequality (24) becomes
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It follows from (24), (25) and (26) that
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proving that the constant
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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.
e-mail:cfx2002@126.com
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