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
• Received : December 10, 2013
• Accepted : July 14, 2014
• Published : September 30, 2014
PDF
e-PUB
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
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
PPT Slide
Lager Image
where
PPT Slide
Lager Image
denotes the integer part of
PPT Slide
Lager Image
and
PPT Slide
Lager Image
For n = 1, we get the midpoint rule
PPT Slide
Lager Image
In [11] , a generalized trapezoid rule is derived by Ujevi ć as follows:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
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,
PPT Slide
Lager Image
where θ ∈ [0, 1] and
PPT Slide
Lager Image
In [5] , Liu established the following generalized perturbed trapezoid rule.
PPT Slide
Lager Image
where Kn ( x ) is the kernel given by
PPT Slide
Lager Image
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
PPT Slide
Lager Image
where
PPT Slide
Lager Image
denotes the integer part of
PPT Slide
Lager Image
and R ( f ) = (−1) n
PPT Slide
Lager Image
PPT Slide
Lager Image
Some sharp perturbed midpoint inequalities are proved by Liu in [6] based on the following identity:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
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
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
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
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
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
Lager Image
where
PPT Slide
Lager Image
denotes the integer part of
PPT Slide
Lager Image
.
Proof . It is not difficult to find the identity
PPT Slide
Lager Image
where Sn ( x ) is the kernel given by
PPT Slide
Lager Image
Using the above identity, we get
PPT Slide
Lager Image
Now, we put
PPT Slide
Lager Image
It is clear that Pn ( x ) and Qn ( x ) are symmetric with respect to the line
PPT Slide
Lager Image
for n even and symmetric with respect to the point
PPT Slide
Lager Image
for n odd. Therefore,
PPT Slide
Lager Image
By substitution
PPT Slide
Lager Image
we find that
PPT Slide
Lager Image
is always negative on [0, 1] for n ≥ 3. Thus
PPT Slide
Lager Image
for n ≥ 3, and
PPT Slide
Lager Image
Hence,
PPT Slide
Lager Image
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:
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Before we end this section, we introduce the notations
PPT Slide
Lager Image
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
PPT Slide
Lager Image
for all x, y ∈ [ a, b ],where L > 0 is given, and, it is said to be ( l,L ) -Lipschitzian on [ a, b ] if
PPT Slide
Lager Image
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
PPT Slide
Lager Image
exists and
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Proof . By (10) and (13) we get
PPT Slide
Lager Image
Then notice that
PPT Slide
Lager Image
Lipschitzian on [ a, b ] and by using (16), we have
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is the usual Lebesgue norm on L 1 [ a, b ].
Proof . By using the identity (10) we get
PPT Slide
Lager Image
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
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is the usual Lebesgue norm on L 2 [ a, b ].
Proof . By using the identity (10) we get
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Proof . By (10) and (13) we get
PPT Slide
Lager Image
then notice that
PPT Slide
Lager Image
a.e. on [ a, b ], we have
PPT Slide
Lager Image
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
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Proof . By (10) and (13) we get
PPT Slide
Lager Image
then notice that f (n) ( x ) − γn ≥ 0 a.e. on [ a, b ], we have
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
where σ(·) is defined by
PPT Slide
Lager Image
Inequality (19) is the best possible in the sense that the constant
PPT Slide
Lager Image
can not be replaced by a smaller one.
Proof . By using the identity (10) and (13) we get
PPT Slide
Lager Image
To prove the sharpness of (19), we suppose that (19) holds with a constant C > 0 as
PPT Slide
Lager Image
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
PPT Slide
Lager Image
It follows that
PPT Slide
Lager Image
Then we can find that the left-hand side of inequality (20) becomes
PPT Slide
Lager Image
and the right-hand side of inequality (20) becomes
PPT Slide
Lager Image
From (20), (21) and (22), we get
PPT Slide
Lager Image
which proving that the constant
PPT Slide
Lager Image
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
PPT Slide
Lager Image
where σ (·) is defined in Theorem 6.1. Inequality (23) is the best possible in the sense that the constant
PPT Slide
Lager Image
can not be replaced by a smaller one.
Proof . By using the identity (10) and (13) we get
PPT Slide
Lager Image
We now suppose that (23) holds with a constant C > 0 as
PPT Slide
Lager Image
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
PPT Slide
Lager Image
It follows that
PPT Slide
Lager Image
Then we can find that the left-hand side of inequality (24) becomes
PPT Slide
Lager Image
and the right-hand side of inequality (24) becomes
PPT Slide
Lager Image
It follows from (24), (25) and (26) that
PPT Slide
Lager Image
proving that the constant
PPT Slide
Lager Image
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
References
Cerone P. , Dragomir S.S. (2000) Trapezoidal type rules from an inequalities point of view, Handbook of Analytic-Computational Methods in Applied Mathematics CRC Press N. Y. 65 - 134
Cerone P. , Dragomir S.S. (2000) Midpoint type rules from an inequalities point of view, Hand-book of Analytic-Computational Methods in Applied Mathematics CRC Press N. Y. 135 - 200
Dragomir S.S. , Agarwal R.P. , Cerone P. (2000) On Simpson’s inequality and applications. J. Inequal. Appl. 5 533 - 579
Liu W.J. , Jiang Y. , Tuna A. 2013 A unified generalization of some quadrature rules and error bounds Appl. Math. Comput. 219 (9) 4765 - 4774    DOI : 10.1016/j.amc.2012.10.093
Liu Z. 2006 Some inequalities of perturbed trapezoid type J. Inequal. in Pure and Appl. Math. Article 47 7 (2)
Liu Z. 2007 A note on perturbed midpoint inequalities Soochow J. Math. 33 (1) 101 - 109
Liu Z. 2011 More on the averaged midpoint-trapezoid type rules Appl. Math. Comput. 218 (4) 1389 - 1398    DOI : 10.1016/j.amc.2011.06.021
Liu Z. 2013 On generalizations of some classical integral inequalities J. Math. Inequal. 7 (2) 255 - 269    DOI : 10.7153/jmi-07-24
Pearce C.E.M. , Pečcarić J. , Ujević N. , Varošsanec S. 2000 Generalizations of some inequalities of Ostrowski-Gruss type Math. Inequal. Appl. 3 (1) 25 - 34
Ujević N. Ujević, 2004 A generalization of Ostrowski’s inequality and applications in numerical inte-gration Appl. Math. Lett. 17 (2) 133 - 137    DOI : 10.1016/S0893-9659(04)90023-7
Ujević N. 2006 Error inequalities for a generalized trapezoid rule Appl. Math. Lett. 19 (1) 32 - 37    DOI : 10.1016/j.aml.2005.03.005
Ujević N. , Erceg G. 2007 A generalization of the corrected midpoint-trapezoid rule and error bounds Appl. Math. Comput. 184 (2) 216 - 222    DOI : 10.1016/j.amc.2006.05.154