Table Of ContentOstrowski Type Inequalities and Applications in
Numerical Integration
Edited By:
Sever S. Dragomir
and
Themistocles M. Rassias
(S.S. Dragomir) School and Communications and Informatics, Victoria
University of Technology, PO Box 14428, Melbourne City, MC 8001,
Victoria, Australia
E-mail address, S.S. Dragomir: [email protected]
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
(T.M.Rassias)DepartmentofMathematics, NationalTechnicalUniver-
sity of Athens, Zografou Campus, 15780 Athens, Greece
1991 Mathematics Subject Classification. Primary 26D15, 26D10; Secondary
41A55, 41A99
Abstract. Ostrowski type inequalities for univariate and multivariate real
functions and their natural applications for numerical quadratures are pre-
sented.
Contents
Preface v
Chapter 1. Generalisations of Ostrowski Inequality and Applications 1
1.1. Introduction 1
1.2. Generalisations for Functions of Bounded Variation 3
1.3. Generalisations for Functions whose Derivatives are in L 15
∞
1.4. Generalisation for Functions whose Derivatives are in L 28
p
1.5. Generalisations in Terms of L −norm 42
1
Bibliography 53
Chapter 2. Integral Inequalities for n−Times Differentiable Mappings 55
2.1. Introduction 55
2.2. Integral Identities 56
2.3. Integral Inequalities 64
2.4. The Convergence of a General Quadrature Formula 71
2.5. Gru¨ss Type Inequalities 75
2.6. Some Particular Integral Inequalities 80
2.7. Applications for Numerical Integration 104
2.8. Concluding Remarks 118
Bibliography 119
Chapter 3. Three Point Quadrature Rules 121
3.1. Introduction 121
3.2. Bounds Involving at most a First Derivative 123
3.3. Bounds for n−Time Differentiable Functions 186
Bibliography 211
Chapter 4. Product Branches of Peano Kernels and Numerical Integration 215
4.1. Introduction 215
4.2. Fundamental Results 217
4.3. Simpson Type Formulae 224
4.4. Perturbed Results 226
4.5. More Perturbed Results Using ∆−Seminorms 235
4.6. Concluding Remarks 241
Bibliography 243
Chapter 5. Ostrowski Type Inequalities for Multiple Integrals 245
5.1. Introduction 245
iii
CONTENTS iv
5.2. An Ostrowski Type Inequality for Double Integrals 249
5.3. Other Ostrowski Type Inequalities 263
5.4. Ostrowski’s Inequality for H¨older Type Functions 273
Bibliography 281
Chapter 6. Some Results for Double Integrals Based on an Ostrowski Type
Inequality 283
6.1. Introduction 283
6.2. The One Dimensional Ostrowski Inequality 284
6.3. Mapping Whose First Derivatives Belong to L (a,b) 284
∞
6.4. Numerical Results 289
6.5. Application For Cubature Formulae 290
6.6. Mapping Whose First Derivatives Belong to L (a,b). 293
p
6.7. Application For Cubature Formulae 296
6.8. Mappings Whose First Derivatives Belong to L (a,b). 298
1
6.9. Integral Identities 301
6.10. Some Integral Inequalities 305
6.11. Applications to Numerical Integration 312
Bibliography 315
Chapter 7. Product Inequalities and Weighted Quadrature 317
7.1. Introduction 317
7.2. Weight Functions 318
7.3. Weighted Interior Point Integral Inequalities 319
7.4. Weighted Boundary Point (Lobatto) Integral Inequalities 332
7.5. Weighted Three Point Integral Inequalities 339
Bibliography 351
Chapter 8. Some Inequalities for Riemann-Stieltjes Integral 353
8.1. Introduction 353
8.2. Some Trapezoid Like Inequalities for Riemann-Stieltjes Integral 355
8.3. Inequalities of Ostrowski Type for the Riemann-Stieltjes Integral 372
8.4. Some Inequalities of Gru¨ss Type for Riemann-Stieltjes Integral 392
Bibliography 401
v
Preface
Itwasnotedintheprefaceofthebook“Inequalities Involving Functions and Their
Integrals and Derivatives”, Kluwer Academic Publishers, 1991, by D.S. Mitrinovi´c,
J.E. Peˇcari´c and A.M. Fink; since the writing of the classical book by Hardy, Lit-
tlewood and Polya (1934), the subject of differential and integral inequalities has
grownbyabout800%. Tenyearson,wecanconfidentlyassertthatthisgrowthwill
increase even more significantly. Twenty pages of Chapter XV in the above men-
tioned book are devoted to integral inequalities involving functions with bounded
derivatives, or, Ostrowski type inequalities. This is now itself a special domain of
the Theory of Inequalities with many powerful results and a large number of ap-
plicationsinNumericalIntegration,ProbabilityTheoryandStatistics,Information
Theory and Integral Operator Theory.
The main aim of this present book, jointly written by the members of the Victoria
University node of RGMIA (Research Group in Mathematical Inequalities and Ap-
plications,http://rgmia.vu.edu.au),istopresentaselectednumberofresultson
Ostrowski type inequalities. Results for univariate and multivariate real functions
andtheirnaturalapplicationsintheerroranalysisofnumericalquadratureforboth
simpleandmultipleintegralsaswellasfortheRiemann-Stieltjesintegralaregiven.
In Chapter 1, authored by S.S. Dragomir and T.M. Rassias, generalisations of the
Ostrowski integralinequality for mappings ofbounded variation and for absolutely
continuousfunctionsviakernelswithn−branchesincludingapplicationsforgeneral
quadrature formulae, are given.
Chapter 2, authored by A. Sofo, builds on the work in Chapter 1. He investigates
generalisations of integral inequalities for n-times differentiable mappings. With
the aid of the modern theory of inequalities and by use of a general Peano kernel,
explicit bounds for interior point rules are obtained. Firstly, he develops inte-
gral equalities which are then used to obtain inequalities for n-times differentiable
mappings on the Lebesgue spaces L [a,b], L [a,b], 1 < p < ∞ and L [a,b]. Sec-
∞ p 1
ondly, some particular inequalities are obtained which include explicit bounds for
perturbed trapezoid, midpoint, Simpson’s, Newton-Cotes, left and right rectangle
rules. Finally, inequalities are also applied to various composite quadrature rules
andtheanalysisallowsthedeterminationofthepartitionrequiredfortheaccuracy
of the result to be within a prescribed error tolerance.
In Chapter 3, authored by P. Cerone and S.S. Dragomir, a unified treatment of
three point quadrature rules is presented in which the classical rules of mid-point,
trapezoidal and Simpson type are recaptured as particular cases. Riemann inte-
grals are approximated for the derivative of the integrand belonging to a variety
of norms. The Gru¨ss inequality and a number of variants are also presented which
provide a variety of inequalities that are suitable for numerical implementation.
Mappings that are of bounded total variation, Lipschitzian and monotonic are also
investigated with relation to Riemann-Stieltjes integrals. Explicit a priori bounds
are provided allowing the determination of the partition required to achieve a pre-
scribed error tolerance.
CONTENTS vi
It is demonstrated that with the above classes of functions, the average of a mid-
point and trapezoidal type rule produces the best bounds.
In Chapter 4, authored by P. Cerone, product branches of Peano kernels are used
to obtain results suitable for numerical integration. In particular, identities and
inequalitiesareobtainedinvolvingevaluationsataninteriorandattheendpoints.
It is shown how previous work and rules in numerical integration are recaptured
as particular instances of the current development. Explicit a priori bounds are
provided allowing the determination of the partition required for achieving a pre-
scribed error tolerance. In the main, Ostrowski-Gru¨ss type inequalities are used to
obtain bounds on the rules in terms of a variety of norms.
In Chapter 5, authored by N.S. Barnett, P. Cerone and S.S. Dragomir, new results
for Ostrowski type inequalities for double and multiple integrals and their applica-
tions for cubature formulae are presented. This work is then continued in Chapter
6,authoredbyG.Hanna,whereanOstrowskitypeinequalityintwodimensionsfor
double integrals on a rectangle region is developed. The resulting integral inequal-
ities are evaluated for the class of functions with bounded first derivative. They
are employed to approximate the double integral by one dimensional integrals and
functionevaluationsusingdifferenttypesofnorms. Iftheone-dimensionalintegrals
are not known, they themselves can be approximated by using a suitable rule, to
produce a cubature rule consisting only of sampling points.
Inaddition,somegeneralisationsofanOstrowskitypeinequalityintwodimensions
for n - time differentiable mappings are given. The result is an integral inequality
with bounded n - time derivatives. This is employed to approximate double inte-
gralsusingonedimensionalintegralsandfunctionevaluationsattheboundaryand
interior points.
In Chapter 7, authored by John Roumeliotis, weighted quadrature rules are inves-
tigated. The results are valid for general weight functions. The robustness of the
bounds is explored for specific weight functions and for a variety of integrands. A
comparison of the current development is made with traditional quadrature rules
and it is demonstrated that the current development has some advantages. In par-
ticular, this method allows the nodes and weights of an n point rule to be easily
obtained,whichmaybepreferentialiftheregionofintegrationvaries. Otherexplicit
errorboundsmaybeobtainedinadvance,thusmakingitpossibletodeterminethe
weight dependent partition required to achieve a certain error tolerance.
In the last chapter, S.S. Dragomir presents recent results in approximating the
Riemann-Stieltjes integral by the use of Trapezoid type, Ostrowski type and Gru¨ss
type inequalities. Applications for certain classes of weighted integrals are also
given.
This book is intended for use in the fields of integral inequalities, approximation
theory, applied mathematics, probability theory and statistics and numerical anal-
ysis.
vii
The Editors,
Melbourne and Athens, December 2000.
CHAPTER 1
Generalisations of Ostrowski Inequality and
Applications
by
S.S. DRAGOMIR and T.M. RASSIAS
Abstract Generalizations of Ostrowski integral inequality for mappings of
bounded variation and for absolutely continuous functions via kernels with
n−branches plus applications for general quadrature formulae are given.
1.1. Introduction
The following result is known in the literature as Ostrowski’s inequality (see for
example [22, p. 468]).
Theorem 1.1. Let f : [a,b] → R be a differentiable mapping on (a,b) with the
property that |f0(t)|≤M for all t∈(a,b). Then
(cid:12)(cid:12) 1 Z b (cid:12)(cid:12) "1 (cid:0)x− a+b(cid:1)2#
(1.1) (cid:12)f(x)− f(t)dt(cid:12)≤ + 2 (b−a)M
(cid:12)(cid:12) b−a a (cid:12)(cid:12) 4 (b−a)2
for all x∈[a,b].
The constant 1 is the best possible in the sense that it cannot be replaced by a
4
smaller constant.
A simple proof of this fact can be done by using the identity:
1 Z b 1 Z b
(1.2) f(x)= f(t)dt+ p(x,t)f0(t)dt, x∈[a,b],
b−a b−a
a a
where
t−a if a≤t≤x
p(x,t):=
t−b if x<t≤b
which also holds for absolutely continuous functions f :[a,b]→R.
The following Ostrowski type result for absolutely continuous functions holds (see
[17], [20] and [18]).
1
1. GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 2
Theorem 1.2. Let f : [a,b] → R be absolutely continuous on [a,b]. Then, for all
x∈[a,b], we have:
(cid:12) (cid:12)
(cid:12) 1 Z b (cid:12)
(1.3) (cid:12)f(x)− f(t)dt(cid:12)
(cid:12) b−a (cid:12)
(cid:12) a (cid:12)
≤ (cid:20)h(p2114+1++1)p1(cid:12)(cid:12)(cid:12)(cid:16)xx(cid:20)−b−(cid:16)b−a−aax+b2a+2−−bbaa(cid:12)(cid:12)(cid:12)(cid:17)i(cid:17)2kp(cid:21)f+(01kb+1−;(cid:16)abb)−−kxaf(cid:17)0kp∞+1(cid:21)p1 (b−a)p1 kf0kq iiff ffp100+∈∈1qLL∞q=[a[1a,,,bb]p],>; 1;
where k·k (r ∈[1,∞]) are the usual Lebesgue norms on L [a,b], i.e.,
r r
kgk :=ess sup |g(t)|
∞
t∈[a,b]
and
!1
Z b r
kgk := |g(t)|rdt , r ∈[1,∞).
r
a
The constants 1, 1 and 1 respectively are sharp in the sense presented in
4 1 2
(p+1)p
Theorem 1.1.
TheaboveinequalitiescanalsobeobtainedfromtheFinkresultin[21]onchoosing
n=1 and performing some appropriate computations.
If one drops the condition of absolute continuity and assumes that f is H¨older
continuous, then one may state the result (see [15])
Theorem 1.3. Let f :[a,b]→R be of r−H−H¨older type, i.e.,
(1.4) |f(x)−f(y)|≤H|x−y|r, for all x,y ∈[a,b],
where r ∈(0,1] and H >0 are fixed. Then for all x∈[a,b] we have the inequality:
(cid:12) (cid:12)
(cid:12) 1 Z b (cid:12)
(1.5) (cid:12)f(x)− f(t)dt(cid:12)
(cid:12) b−a (cid:12)
(cid:12) a (cid:12)
H "(cid:18)b−x(cid:19)r+1 (cid:18)x−a(cid:19)r+1#
≤ + (b−a)r.
r+1 b−a b−a
The constant 1 is also sharp in the above sense.
r+1
Note that if r =1, i.e., f is Lipschitz continuous, then we get the following version
of Ostrowski’s inequality for Lipschitzian functions (with L instead of H) due to
S.S. Dragomir ([13], see also [2])
(cid:12) (cid:12) !2
(cid:12) 1 Z b (cid:12) 1 x− a+b
(1.6) (cid:12)(cid:12)f(x)− b−a f(t)dt(cid:12)(cid:12)≤4 + b−a2 (b−a)L.
(cid:12) a (cid:12)
Here the constant 1 is also best.
4
