Cerone / Dragomir Mathematical Inequalities

A Perspective
1. Auflage 2011
ISBN: 978-1-4398-4897-5
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: 0 - No protection

E-Book, Englisch, 391 Seiten

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 include numerous classical and recent results that are comprehensible to both experts and general scientists. They describe key inequalities for real or complex numbers and sequences in analysis, including the Abel; the Biernacki, Pidek, and Ryll–Nardzewski; Cebysev’s; the Cauchy–Bunyakovsky–Schwarz; and De Bruijn’s inequalities. They also focus on the role of integral inequalities, such as Hermite–Hadamard inequalities, in modern analysis. In addition, the book covers Schwarz, Bessel, Boas–Bellman, Bombieri, Kurepa, Buzano, Precupanu, Dunkl–William, and Grüss inequalities as well as generalizations of Hermite–Hadamard inequalities for isotonic linear and sublinear functionals.

For each inequality presented, results are complemented with many unique remarks that reveal rich interconnections between the inequalities. These discussions create a natural platform for further research in applications and related fields.

Zielgruppe

Researchers and graduate students in real and functional analysis.

Autoren/Hrsg.

Cerone, Pietro

Dragomir, Silvestru Sever

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

Discrete Inequalities
An Elementary Inequality for Two Numbers
An Elementary Inequality for Three Numbers
A Weighted Inequality for Two Numbers
The Abel Inequality
The Biernacki, Pidek, and Ryll–Nardzewski (BPR) Inequality
Cebysev’s Inequality for Synchronous Sequences
The Cauchy–Bunyakovsky–Schwarz (CBS) Inequality for Real Numbers
The Andrica–Badea Inequality
A Weighted Grüss-Type Inequality
Andrica–Badea’s Refinement of the Grüss Inequality
Cebysev-Type Inequalities
De Bruijn’s Inequality
Daykin–Eliezer–Carlitz’s Inequality
Wagner’s Inequality
The Pólya–Szegö Inequality
The Cassels Inequality
Hölder’s Inequality for Sequences of Real Numbers
The Minkowski Inequality for Sequences of Real Numbers
Jensen’s Discrete Inequality
A Converse of Jensen’s Inequality for Differentiable Mappings
The Petrovic Inequality for Convex Functions
Bounds for the Jensen Functional in Terms of the Second Derivative
Slater’s Inequality for Convex Functions
A Jensen-Type Inequality for Double Sums
Integral Inequalities for Convex Functions
The Hermite–Hadamard Integral Inequality
Hermite–Hadamard Related Inequalities
Hermite–Hadamard Inequality for Log-Convex Mappings
Hermite–Hadamard Inequality for the Godnova–Levin Class of Functions
The Hermite–Hadamard Inequality for Quasi-Convex Functions
The Hermite–Hadamard Inequality for s-Convex Functions in the Orlicz Sense
The Hermite–Hadamard Inequality for s-Convex Functions in the Breckner Sense
Inequalities for Hadamard’s Inferior and Superior Sums
A Refinement of the Hermite–Hadamard Inequality for the Modulus
Ostrowski and Trapezoid-Type Inequalities
Ostrowski’s Integral Inequality for Absolutely Continuous Mappings
Ostrowski’s Integral Inequality for Mappings of Bounded Variation
Trapezoid Inequality for Functions of Bounded Variation
Trapezoid Inequality for Monotonic Mappings
Trapezoid Inequality for Absolutely Continuous Mappings
Trapezoid Inequality in Terms of Second Derivatives
Generalised Trapezoid Rule Involving nth Derivative Error Bounds
A Refinement of Ostrowski’s Inequality for the Cebysev Functional
Ostrowski-Type Inequality with End Interval Means
Multidimensional Integration via Ostrowski Dimension Reduction
Multidimensional Integration via Trapezoid and Three Point
Generators with Dimension Reduction
Relationships between Ostrowski, Trapezoidal, and Cebysev Functionals
Perturbed Trapezoidal and Midpoint Rules
A Cebysev Functional and Some Ramifications
Weighted Three Point Quadrature Rules
Grüss-Type Inequalities and Related Results
The Grüss Integral Inequality
The Grüss–Cebysev Integral Inequality
Karamata’s Inequality
Steffensen’s Inequality
Young’s Inequality
Grüss-Type Inequalities for the Stieltjes Integral of Bounded Integrands
Grüss-Type Inequalities for the Stieltjes Integral of Lipschitzian Integrands
Other Grüss-Type Inequalities for the Riemann–Stieltjes Integral
Inequalities for Monotonic Integrators
Generalisations of Steffensen’s Inequality over Subintervals
Inequalities in Inner Product Spaces
Schwarz’s Inequality in Inner Product Spaces
A Conditional Refinement of the Schwarz Inequality
The Duality Schwarz-Triangle Inequalities
A Quadratic Reverse for the Schwarz Inequality
A Reverse of the Simple Schwarz Inequality
A Reverse of Bessel’s Inequality
Reverses for the Triangle Inequality in Inner Product Spaces
The Boas–Bellman Inequality
The Bombieri Inequality
Kurepa’s Inequality
Buzano’s Inequality
A Generalisation of Buzano’s Inequality
Generalisations of Precupanu’s Inequality
The Dunkl–William Inequality
The Grüss Inequality in Inner Product Spaces
A Refinement of the Grüss Inequality in Inner Product Spaces
Inequalities in Normed Linear Spaces and for Functionals
A Multiplicative Reverse for the Continuous Triangle Inequality
Additive Reverses for the Continuous Triangle Inequality
Reverses of the Discrete Triangle Inequality in Normed Spaces
Other Multiplicative Reverses for a Finite Sequence of Functionals
The Diaz–Metcalf Inequality for Semi-Inner Products
Multiplicative Reverses of the Continuous Triangle Inequality
Reverses in Terms of a Finite Sequence of Functionals
Generalisations of the Hermite–Hadamard Inequalities for Isotonic Linear Functionals
A Symmetric Generalisation
Generalisations of the Hermite–Hadamard Inequality for Isotonic Sublinear Functionals
References
Index

Über Autor(innen)

Pietro Cerone is a professor of mathematics at Victoria University, where he served as head of the School of Computer Science and Mathematics from 2003 to 2008. Dr. Cerone is on the editorial board of a dozen international journals and has published roughly 200 refereed works in the field. His research interests include mathematical modeling, population dynamics, and applications of mathematical inequalities.
Sever S. Dragomir is a professor of mathematics and chair of the international Research Group in Mathematical Inequalities and Applications at Victoria University. Dr. Dragomir is an editorial board member of more than 30 international journals and has published over 600 research articles. His research in pure and applied mathematics encompasses classical mathematical analysis, operator theory, Banach spaces, coding, adaptive quadrature and cubature rules, differential equations, and game theory.