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New Perspectives on the Theory of Inequalities for Integral and Sum

Author: Nazia Irshad

Publisher: Springer Nature

Published: 2022-03-29

Total Pages: 319

ISBN-13: 3030905632

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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.

Book Synopsis New Perspectives on the Theory of Inequalities for Integral and Sum by : Nazia Irshad

Download or read book New Perspectives on the Theory of Inequalities for Integral and Sum written by Nazia Irshad and published by Springer Nature. This book was released on 2022-03-29 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Author: Nassif Ghoussoub

Publisher: American Mathematical Soc.

Published: 2013-04-09

Total Pages: 331

ISBN-13: 0821891529

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"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.

Book Synopsis Functional Inequalities: New Perspectives and New Applications by : Nassif Ghoussoub

Download or read book Functional Inequalities: New Perspectives and New Applications written by Nassif Ghoussoub and published by American Mathematical Soc.. This book was released on 2013-04-09 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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

Author: Pietro Cerone

Publisher: CRC Press

Published: 2010-12-01

Total Pages: 391

ISBN-13: 1439848971

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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

Book Synopsis Mathematical Inequalities by : Pietro Cerone

Download or read book Mathematical Inequalities written by Pietro Cerone and published by CRC Press. This book was released on 2010-12-01 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author:

Publisher: Nova Publishers

Published: 2007

Total Pages: 202

ISBN-13: 9781594548758

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Book Synopsis Inequality Theory and Applications. by :

Download or read book Inequality Theory and Applications. written by and published by Nova Publishers. This book was released on 2007 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt:

DIFFERENTIAL AND INTEGRAL INEQUALITIES

Author: V. LAKKSHMIKANTHAM

Publisher:

Published:

Total Pages:

ISBN-13:

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Book Synopsis DIFFERENTIAL AND INTEGRAL INEQUALITIES by : V. LAKKSHMIKANTHAM

Download or read book DIFFERENTIAL AND INTEGRAL INEQUALITIES written by V. LAKKSHMIKANTHAM and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Hilbert-Type Integral Inequalities

Author: Bicheng Yang

Publisher:

Published: 2010-04-02

Total Pages: 132

ISBN-13: 9781608050338

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Hilbert-type integral inequalities, including the well known Hilbert's integral inequality published in 1908, are important in analysis and its applications. This well organized handbook covers the newest methods of weight functions and most important recent results of Hilbert-type integral inequalities and applications in three classes of normal spaces. It is clear and well written, suitable for researchers, mathematicians and advanced students who wish to increase their familiarity with different topics of modern and classical mathematical inequalities related to Real Analysis and Operator Theory. Both basic and applied aspects are presented in this book. The many illustrative worked-out examples given in several chapters should help readers acquire expertise in this field.

Book Synopsis Hilbert-Type Integral Inequalities by : Bicheng Yang

Download or read book Hilbert-Type Integral Inequalities written by Bicheng Yang and published by . This book was released on 2010-04-02 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hilbert-type integral inequalities, including the well known Hilbert's integral inequality published in 1908, are important in analysis and its applications. This well organized handbook covers the newest methods of weight functions and most important recent results of Hilbert-type integral inequalities and applications in three classes of normal spaces. It is clear and well written, suitable for researchers, mathematicians and advanced students who wish to increase their familiarity with different topics of modern and classical mathematical inequalities related to Real Analysis and Operator Theory. Both basic and applied aspects are presented in this book. The many illustrative worked-out examples given in several chapters should help readers acquire expertise in this field.

Measure Theory

Author: Vladimir I. Bogachev

Publisher: Springer Science & Business Media

Published: 2007-01-15

Total Pages: 1075

ISBN-13: 3540345140

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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.

Book Synopsis Measure Theory by : Vladimir I. Bogachev

Download or read book Measure Theory written by Vladimir I. Bogachev and published by Springer Science & Business Media. This book was released on 2007-01-15 with total page 1075 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Author: Alberto Torchinsky

Publisher: Springer Nature

Published: 2022-10-05

Total Pages: 182

ISBN-13: 3031117999

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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.

Book Synopsis A Modern View of the Riemann Integral by : Alberto Torchinsky

Download or read book A Modern View of the Riemann Integral written by Alberto Torchinsky and published by Springer Nature. This book was released on 2022-10-05 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Means and Their Inequalities

Author: P.S. Bullen

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 476

ISBN-13: 9401722269

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Approach your problems from the right end It isn't !hat they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal 0/ Fa/her 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van GuJik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fie1ds does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "complete1y integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing c1assification schemes. They draw upon wide1y different sections of mathematics.

Book Synopsis Means and Their Inequalities by : P.S. Bullen

Download or read book Means and Their Inequalities written by P.S. Bullen and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 476 pages. Available in PDF, EPUB and Kindle. Book excerpt: Approach your problems from the right end It isn't !hat they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal 0/ Fa/her 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van GuJik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fie1ds does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "complete1y integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing c1assification schemes. They draw upon wide1y different sections of mathematics.

A Comprehensive Treatment of q-Calculus

Author: Thomas Ernst

Publisher: Springer Science & Business Media

Published: 2012-09-08

Total Pages: 491

ISBN-13: 303480430X

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To date, the theoretical development of q-calculus has rested on a non-uniform basis. Generally, the bulky Gasper-Rahman notation was used, but the published works on q-calculus looked different depending on where and by whom they were written. This confusion of tongues not only complicated the theoretical development but also contributed to q-calculus remaining a neglected mathematical field. This book overcomes these problems by introducing a new and interesting notation for q-calculus based on logarithms.For instance, q-hypergeometric functions are now visually clear and easy to trace back to their hypergeometric parents. With this new notation it is also easy to see the connection between q-hypergeometric functions and the q-gamma function, something that until now has been overlooked. The book covers many topics on q-calculus, including special functions, combinatorics, and q-difference equations. Apart from a thorough review of the historical development of q-calculus, this book also presents the domains of modern physics for which q-calculus is applicable, such as particle physics and supersymmetry, to name just a few.

Book Synopsis A Comprehensive Treatment of q-Calculus by : Thomas Ernst

Download or read book A Comprehensive Treatment of q-Calculus written by Thomas Ernst and published by Springer Science & Business Media. This book was released on 2012-09-08 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: To date, the theoretical development of q-calculus has rested on a non-uniform basis. Generally, the bulky Gasper-Rahman notation was used, but the published works on q-calculus looked different depending on where and by whom they were written. This confusion of tongues not only complicated the theoretical development but also contributed to q-calculus remaining a neglected mathematical field. This book overcomes these problems by introducing a new and interesting notation for q-calculus based on logarithms.For instance, q-hypergeometric functions are now visually clear and easy to trace back to their hypergeometric parents. With this new notation it is also easy to see the connection between q-hypergeometric functions and the q-gamma function, something that until now has been overlooked. The book covers many topics on q-calculus, including special functions, combinatorics, and q-difference equations. Apart from a thorough review of the historical development of q-calculus, this book also presents the domains of modern physics for which q-calculus is applicable, such as particle physics and supersymmetry, to name just a few.