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A chapter in the book "Inequalities Involving Functions and Their Integrals and
Derivatives", Kluwer Academic Publishers, 1991, by Mitrinovic,
Pecaric and Fink is devoted to integral inequalities involving functions with bounded
derivatives, or, Ostrowski type inequalities. This topic has now become a special domain in the Theory of Inequalities, there
having been published many powerful results and a large number of applications in Numerical Integration, Probability Theory and
Statistics, Information Theory and Integral Operator Theory.
The first monograph devoted to Ostrowski type inequalities and applications for quadrature rules was written by members of the
Research Group in Mathematical Inequalities and Applications (RGMIA, see http://rgmia.vu.edu.au) in 2002. The book was entitled
``Ostrowski Type Inequalities and Applications in Numerical
Integration'', edited by S.S. Dragomir & Th. M. Rassias, Kluwer Academic Publishers. The main aim of this
monograph was to present some selected results of Ostrowski type inequalities for univariate and multivariate real functions and
their natural application to the error analysis of numerical quadrature for both simple and multiple integrals as well as for
the Riemann-Stieltjes integral. Due to space limitations, however, no attempt was made to present
applications in other domains, more specifically, in Probability Theory.
It can be observed that Ostrowski type inequalities may also be successfully used to obtain various tight bounds for the
expectation, variance and moments of continuous random variables defined over a finite interval. This had been noted in the late
1990's by many authors including members of the RGMIA located at Victoria University, Melbourne, Australia (see for instance the
RGMIA Res. Rep. Coll., http://ajmaa.org/RGMIA/issues.php for the years 1998-1999). The domain is now rich with results whose
beneficial value will increase by being presented in a unified manner. This will then provide to all interested in Inequalities
in Applied Probability Theory & Statistics, a primer of results and techniques that may well need further attention and polishing
so as to obtain the best possible bounds and estimates. It is from this view point that the current book is written and it is
intended to be useful to both graduate students and established researchers working in Probability Theory
& Statistics, Analytic Integral Inequalities and their applications in demography,
economics, physics, biology, and other scientific areas.
The chapter outlines are given below and it is intended that they can be
read independently if desired. The first two chapters are concerned with natural applications to
cumulative distribution functions (CDFs) and expectations for random variables (RVs) over a finite interval. The results use the
latest Ostrowski type integral inequalities for functions that are of: bounded variation, convex,
Hölder continuous, Lipschitzian or absolutely continuous. The tools used are both the
Riemann-Stieltjes integral and the Lebesgue integral. Chapter 3 investigates the use of trapezoidal or
corrected trapezoidal type inequalities developed recently in parallel with
Ostrowski type inequalities for various classes of functions including the
ones mentioned previously, but also for classes of much smoother functions
whose second, third or fourth derivatives belong to the Lebesgue spaces Lp for
p = 1. Chapter 4, deals with Grüss type or pre-Grüss type integral inequalities which provide error
bounds for approximating the integral mean of a product (of two functions) in terms of the product of the integral
means (for each individual function). Such inequalities are useful when the integral means of the individual functions are known or
are more convenient to calculate. They also provide more accurate approximations, since the bounds are expressed in terms of the
oscillation of a function rather than its sup norm that is usually not as tight. Utilising this type of estimate, various bounds for
mathematical expressions incorporating the CDFs and the expectations are provided. Elementary and simple-looking bounds
for the variance of continuous RVs are presented in Chapter 5. The tools used here are mostly
Grüss and pre-Grüss type inequalities and some recent results obtained by the authors in connection with the problem of
bounding the Cebysev functional in its integral version over finite intervals in terms of various quantities and under
certain assumptions for the involved integrable functions. Finally, in Chapter 6, by employing Taylor type expansions for
$n$-time differentiable CDFs, various bounds involving the variance of a
continuous random variable defined on a finite interval that are more accurate in terms of order of convergence, are outlined.
The book is self-contained in the sense that the reader needs only to be
familiar with basic real analysis, integration theory and probability theory. All inequalities used in the text are explicitly stated and
appropriately referenced. A comprehensive list of references on which the
book is based is presented, complemented by other relevant literature that
will allow the interested reader to be introduced to open problems, including the necessity to extend some of the obtained results to
probability density functions defined on unbounded intervals.
Last, but no means least, the authors would like especially thank Professor George Anastassiou from Memphis University for his
constant encouragement to write the book and whose numerous comments have been implemented in the final version.
The Authors
Melbourne, November 2004
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P. Cerone