Keywords (Extracted from title, table of contents and
abstract of thesis) Ostrowski, Grüss inequality,
inequality, Numerical, Integration, Special, Means, Random variable,
Probability, Density, Cumulative, Distribution, Function, Nonlinear,
Equations, Iterative, Methods, Generalizations, applications, 
Abstract 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 functions namely, inequalities of Grüss [46]
and µCebyšev type (see [64], p. 297).
These 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 applications of inequalities of OstrowskiGrüss and µCebyšev
type were established.
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 nonlinear 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 Lp (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 LLipschitzian 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 LLipschitzian functions and (l;L)Lipschitzian
functions.
The …rst inequality of OstrowskiGrüss 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 L2 (a;
b). Some generalizations of OstrowskiGrüss 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. In the recent past,
many researchers have used µCebyšev type functionals to
obtain some new product inequalities of Ostrowski, µCebyšev, 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 Lp (a; b) ; p > 1 and
for twice di¤erentiable functions in L1 (a; b .A product inequality
is also given for monotonic nondecreasing functions. The
inequalities are then applied to the expectation of a random
variable.
In [3], G. A. Anastassiou has extended µCebyševGrüss type
inequalities on RN over spherical shells and balls. We have extended
this inequality for ndimensional Euclidean space over spherical
shells and balls on Lp [a; b] ; p > 1. Some weighted Ostrowski type
inequalities for a random variable whose proba bility density
functions belong to fLp (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 fLp (a; b) ; p = 1; p > 1g.This
dissertation also describes the applications of specially derived
Ostrowski type inequalities to obtain some twostep and threestep
iterative methods for solving nonlinear equations.
Some Ostrowski type inequalities for NewtonCotes formulae are also
presented in a generalized or optimal manner to obtain one point,
twopoint and fourpoint NewtonCotes formulae of open as well as
closed type.The results presented here extend various inequalities
of Ostrowski type upto their year of publication.
