A Survey on Cauchy-Buniakowsky-Schwartz Type Discrete Inequalities
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|The Cauchy-Buniakowski-Schwartz inequality, or for short, the (CBS)- inequality, plays an important role in different branches of Modern Mathematics including Hilbert Spaces Theory, Probability
& Statistics, Classical Real and Complex Analysis, Numerical Analysis, Qualitative Theory of Differential Equations and their applications.
The main purpose of this survey is to identify and highlight the discrete inequalities that are connected with (CBS)- inequality and provide refinements, counterparts and reverse results as well as to study some functional properties of certain mappings that can be naturally associated with this inequality such as superadditivity, supermultiplicity, the strong versions of these and the corresponding monotonicity properties. Many companions and related results both for real and complex numbers are also presented.
The first chapter is devoted to a number of (CBS)- type inequalities that provides not only natural generalizations but also several extensions for different classes of analytic functions of a real variable. A generalization of the Wagner inequality for complex numbers is obtained. Several results discovered by the author in the late eighties and published in different journals of lesser circulation are also surveyed.
The second chapter contains different refinements of the (CBS)- inequality including de Bruijn's inequality, McLaughlin's inequality, the Daykin-Eliezer-Carlitz result in the version presented by Mitrinovic-Pecaric and Fink as well as the refinements of a particular version obtained by Alzer and Zheng. A number of new results obtained by the author, which are connected with the above ones, are also presented.
Chapter 3 is devoted to the study of functional properties of different mappings naturally associated to the (CBS)- inequality. Properties such as superadditivity, strong superadditivity, monotonicity and supermultiplicity and the corresponding inequalities are mentioned.
In the next chapter, Chapter 4, counterpart results for the (CBS)- inequality are surveyed. The results of Cassels, Pólya-Szegö, Greub-Rheinbold, Shisha-Mond and Zagier are presented with their original proofs. New results and versions for complex numbers are also obtained. Counterparts in terms of p-norms of the forward difference recently discovered by the author and some refinements of Cassels and Pólya-Szegö results obtained via Andrica-Badea inequality are mentioned. Some new facts derived from Grüss type inequalities are also pointed out.
Chapter 5 is devoted to various inequalities related to the (CBS)- inequality. The two inequalities obtained by Ostrowski and Fan-Todd results are presented. New inequalities obtained via Jensen type inequality for convex functions are derived, some inequalities for the Cebysev functionals are pointed out. Versions for complex numbers that generalize Ostrowski results are also emphasised.
It was one of the main aims of the survey to provide complete proofs for the results considered. We also note that in most cases only the original references are mentioned.
Being self contained, the survey may be used by both postgraduate students and researchers interested in Theory of Inequalities & Applications as well as by Mathematicians and other Scientists dealing with numerical computations, bounds and estimates where the (CBS)- inequality may be used as a powerful tool.
The author intends to continue this survey with another one devoted to the functional and integral versions of the (CBS)- inequality. The corresponding results holding in inner-product and normed spaces will be considered as well.
To reference this book, please use the following:
Alpha version: 20th January, 2000.