dc.description.abstract |
Some Generalization of Ostrowski Inequalities with Applications
in Numerical Integration and Special Means
by
Fiza Zafar
Submitted to the Centre for Advanced Studies in Pure and Applied
Mathematics, Bahauddin Zakariya University, Multan on March 05, 2009
in partial ful...llment of the requirement for the degree of
Doctor in Philosophy of Mathematics.
μ
Keywords: Ostrowski inequality, Grüss inequality, Cebyš
ev inequality, Numer-
ical Integration, Special Means, Random variable, Probability Density Functions,
Cumulative Distribution Function, Nonlinear Equations, Iterative Methods.
2000 Mathematics Subject Classi...cation: 26D10; 26D15; 26D20; 41A55;
60E15; 34A34; 26C10; 65H05.
In the last few decades, the ...eld of mathematical inequalities has proved to be
an extensively applicable ...eld. It is applicable in the following manner:
Integral inequalities play an important role in several other branches of math-
ematics and statistics with reference to its applications.
The elementary inequalities are proved to be helpful in the development of
many other branches of mathematics.
The development of inequalities has been established with the publication of the
books by G. H. Hardy, J. E. Littlewood and G. Polya [47] in 1934, E. F. Beckenbach
and R. Bellman [13] in 1961 and by D. S. Mitrinovi ́c, J. E. Peμcari ́c and A. M. Fink
[64] & [65] in 1991. The publication of later has resulted to bring forward some
new integral inequalities involving functions with bounded derivatives that measure
bounds on the deviation of functional value from its mean value namely, Ostrowski
inequality [69]. The books by D. S. Mitrinovi ́c, J. E. Peμcari ́c and A. M. Fink have
also brought to focus integral inequalities which establish a connection between the
integral of the product of two functions and the product of the integrals of the two
μ
functions namely, inequalities of Grüss [46] and Cebyš
ev type (see [64], p. 297).
iiiThese type of inequalities are of supreme importance because they have immediate
applications in Numerical integration, Probability theory, Information theory and
Integral operator theory. The monographs presented by S. S. Dragomir and Th.
M. Rassias [36] in 2002 and by N. S. Barnett, P. Cerone and S. S. Dragomir [8]
in 2004 can well justify this statement. In these monographs, separate aspects of
μ
applications of inequalities of Ostrowski-Grüss and Cebyš
ev type were established.
The main aim of this dissertation is to address the domains of establishing
μ
inequalities of Ostrowski-Grüss and Cebyš
ev type and their applications in Statis-
tics, Numerical integration and Non-linear analysis. The tools that are used are
Peano kernel approach, the most classical and extensively used approach in devel-
oping such integral inequalities, Lebesgue and Riemann-Stieltjes integrals, Lebesgue
μ
spaces, Korkine’s identity [52], the classical Cebyš
ev functional, Pre-Grüss and Pre-
μ
Cebyš
ev inequalities proved in [60].
This dissertation presents some generalized Ostrowski type inequalities. These
inequalities are being presented for nearly all types of functions i.e., for higher
di¤erentiable functions, bounded functions, absolutely continuous functions, (l; L)-
Lipschitzian functions, monotonic functions and functions of bounded variations.
The inequalities are then applied to composite quadrature rules, special means,
probability density functions, expectation of a random variable, beta random vari-
able and to construct iterative methods for solving non-linear equations.
The generalizations to the inequalities are obtained by introducing arbitrary
parameters in the Peano kernels involved. The parameters can be so adjusted to
recapture the previous results as well as to obtain some new estimates of such
inequalities.
The Ostrowski type inequalities for twice di¤erentiable functions have been ex-
tensively addressed by N. S. Barnett et al. and Zheng Liu in [9] and [59]. We have
presented some perturbed inequalities of Ostrowski type in L p (a; b) ; p
1; p = 1
which generalize and re...ne the results of [9] and [59].
In the past few years, Ostrowski type inequalities are developed for functions
in higher spaces i.e., for L-Lipschitzian functions and (l; L)-Lipschitzian functions.
We, in here, have obtained Ostrowski type inequality for n- di¤erentiable (l; L)-
Lipschitzian functions, a generalizations of such inequalities for L-Lipschitzian func-
ivtions and (l; L)-Lipschitzian functions.
The ...rst inequality of Ostrowski-Grüss type was presented by S. S. Dragomir
and S. Wang in [39]. In this dissertation, some improved and generalized Ostrowski-
Grüss type inequalities are further generalized for the ...rst and twice di¤erentiable
functions in L 2 (a; b). Some generalizations of Ostrowski-Grüss type inequality in
terms of upper and lower bounds of the ...rst and twice di¤erentiable functions are
also given. The inequalities are then applied to probability density functions, special
means, generalized beta random variable and composite quadrature rules.
μ
In the recent past, many researchers have used Cebyš
ev type functionals to
μ
obtain some new product inequalities of Ostrowski-, Cebyš
ev-, and Grüss type. We,
in here, have also taken into account this domain to present some generalizations
and improvements of such inequalities. The generalizations are obtained for ...rst
di¤erentiable absolutely continuous functions with ...rst derivatives in L p (a; b) ; p >
1 and for twice di¤erentiable functions in L 1 (a; b). A product inequality is also
given for monotonic non-decreasing functions. The inequalities are then applied to
the expectation of a random variable.
μ
In [3], G. A. Anastassiou has extended Cebyš
ev-Grüss type inequalities on R N
over spherical shells and balls. We have extended this inequality for n-dimensional
Euclidean space over spherical shells and balls on L p [a; b] ; p > 1.
Some weighted Ostrowski type inequalities for a random variable whose proba-
bility density functions belong to fL p (a; b) ; p = 1; p > 1g are presented as weighted
extensions of the results of [10] and [33]. Ostrowski type inequalities are also applied
to obtain various tight bounds for the random variables de...ned on a ...nite intervals
whose probability density functions belong to fL p (a; b) ; p = 1; p > 1g.
This dissertation also describes the applications of specially derived Ostrowski
type inequalities to obtain some two-step and three-step iterative methods for solv-
ing non-linear equations.
Some Ostrowski type inequalities for Newton-Cotes formulae are also presented
in a generalized or optimal manner to obtain one-point, two-point and four-point
Newton-Cotes formulae of open as well as closed type.
The results presented here extend various inequalities of Ostrowski type upto
their year of publication. |
en_US |