New Perspectives on the Theory of Inequalities for Integral and Sum
Title | New Perspectives on the Theory of Inequalities for Integral and Sum PDF eBook |
Author | Nazia Irshad |
Publisher | Springer Nature |
Pages | 319 |
Release | 2022-03-29 |
Genre | Mathematics |
ISBN | 3030905632 |
This book provides new contributions to the theory of inequalities for integral and sum, and includes four chapters. In the first chapter, linear inequalities via interpolation polynomials and green functions are discussed. New results related to Popoviciu type linear inequalities via extension of the Montgomery identity, the Taylor formula, Abel-Gontscharoff's interpolation polynomials, Hermite interpolation polynomials and the Fink identity with Green’s functions, are presented. The second chapter is dedicated to Ostrowski’s inequality and results with applications to numerical integration and probability theory. The third chapter deals with results involving functions with nondecreasing increments. Real life applications are discussed, as well as and connection of functions with nondecreasing increments together with many important concepts including arithmetic integral mean, wright convex functions, convex functions, nabla-convex functions, Jensen m-convex functions, m-convex functions, m-nabla-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator of order m. The fourth chapter is mainly based on Popoviciu and Cebysev-Popoviciu type identities and inequalities. In this last chapter, the authors present results by using delta and nabla operators of higher order.
Functional Inequalities: New Perspectives and New Applications
Title | Functional Inequalities: New Perspectives and New Applications PDF eBook |
Author | Nassif Ghoussoub |
Publisher | American Mathematical Soc. |
Pages | 331 |
Release | 2013-04-09 |
Genre | Mathematics |
ISBN | 0821891529 |
"The book describes how functional inequalities are often manifestations of natural mathematical structures and physical phenomena, and how a few general principles validate large classes of analytic/geometric inequalities, old and new. This point of view leads to "systematic" approaches for proving the most basic inequalities, but also for improving them, and for devising new ones--sometimes at will and often on demand. These general principles also offer novel ways for estimating best constants and for deciding whether these are attained in appropriate function spaces. As such, improvements of Hardy and Hardy-Rellich type inequalities involving radially symmetric weights are variational manifestations of Sturm's theory on the oscillatory behavior of certain ordinary differential equations. On the other hand, most geometric inequalities, including those of Sobolev and Log-Sobolev type, are simply expressions of the convexity of certain free energy functionals along the geodesics on the Wasserstein manifold of probability measures equipped with the optimal mass transport metric. Caffarelli-Kohn-Nirenberg and Hardy-Rellich-Sobolev type inequalities are then obtained by interpolating the above two classes of inequalities via the classical ones of Hölder. The subtle Moser-Onofri-Aubin inequalities on the two-dimensional sphere are connected to Liouville type theorems for planar mean field equations."--Publisher's website.
Mathematical Inequalities
Title | Mathematical Inequalities PDF eBook |
Author | Pietro Cerone |
Publisher | CRC Press |
Pages | 391 |
Release | 2010-12-01 |
Genre | Mathematics |
ISBN | 1439848971 |
Drawing on the authors' research work from the last ten years, Mathematical Inequalities: A Perspective gives readers a different viewpoint of the field. It discusses the importance of various mathematical inequalities in contemporary mathematics and how these inequalities are used in different applications, such as scientific modeling.The authors
Inequality Theory and Applications.
Title | Inequality Theory and Applications. PDF eBook |
Author | |
Publisher | Nova Publishers |
Pages | 202 |
Release | 2007 |
Genre | Inequalities (Mathematics) |
ISBN | 9781594548758 |
Convexity from the Geometric Point of View
Title | Convexity from the Geometric Point of View PDF eBook |
Author | Vitor Balestro |
Publisher | Springer Nature |
Pages | 1195 |
Release | |
Genre | |
ISBN | 3031505077 |
Measure Theory
Title | Measure Theory PDF eBook |
Author | Vladimir I. Bogachev |
Publisher | Springer Science & Business Media |
Pages | 1075 |
Release | 2007-01-15 |
Genre | Mathematics |
ISBN | 3540345140 |
This book giving an exposition of the foundations of modern measure theory offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course, and, finally, more specialized topics partly covered by more than 850 exercises with detailed hints and references. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided.
A Modern View of the Riemann Integral
Title | A Modern View of the Riemann Integral PDF eBook |
Author | Alberto Torchinsky |
Publisher | Springer Nature |
Pages | 182 |
Release | 2022-10-05 |
Genre | Mathematics |
ISBN | 3031117999 |
This monograph uncovers the full capabilities of the Riemann integral. Setting aside all notions from Lebesgue’s theory, the author embarks on an exploration rooted in Riemann’s original viewpoint. On this journey, we encounter new results, numerous historical vignettes, and discover a particular handiness for computations and applications. This approach rests on three basic observations. First, a Riemann integrability criterion in terms of oscillations, which is a quantitative formulation of the fact that Riemann integrable functions are continuous a.e. with respect to the Lebesgue measure. Second, the introduction of the concepts of admissible families of partitions and modified Riemann sums. Finally, the fact that most numerical quadrature rules make use of carefully chosen Riemann sums, which makes the Riemann integral, be it proper or improper, most appropriate for this endeavor. A Modern View of the Riemann Integral is intended for enthusiasts keen to explore the potential of Riemann's original notion of integral. The only formal prerequisite is a proof-based familiarity with the Riemann integral, though readers will also need to draw upon mathematical maturity and a scholarly outlook.