The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences. It is freely available on the Web (http://www.research.att.com/~njas/sequences/) and is widely used.

There are several ways in which it benefits research: 1 It serves as a dictionary, to tell the user what is known about a particular sequence. There are hundreds of papers which thank the OEIS for assistance in this way.
 1 The associated Sequence Fans mailing list is a worldwide network which has evolved into a powerful machine for tackling new problems.
 1 As a direct source of new theorems, when a sequence arises in two different contexts.
 1 As a source of new research, when one sees a sequence in the OEIS that cries out to be analyzed.
 

The 40-year history of the OEIS recapitulates the story of modern computing, from punched cards to the internet.

The talk will be illustrated with numerous examples, emphasizing new sequences that have arrived in the past few months. Many open problems will be mentioned.

Because of the profusion of books and journals, volunteers play an important role in maintaining the database. If you come across an interesting number sequence in a book, journal or web site, please send it and the reference to the OEIS. (You do not need to be the author of the sequence to do this.) There is a web site for sending in "Comments" or "New sequences".

Several new features have been added to the OEIS in the past year. Thanks to the work of Russ Cox, searches are now performed at high speed, and thanks to the work of Debby Swayne, there is a button which displays plots of each sequence. Finally, a "listen" button enables one to hear the sequence played on a musical instrument (try Recamaan's sequence A005132!).

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The On-Line Encyclopedia of Integer Sequences

Introduction to linear and nonlinear programming

Preface to 'Means and their Inequalities'. Preface to the Handbook. Basic References. Notations. 1. Referencing. 2. Bibliographic References. 3. Symbols for some Important Inequalities. 4. Numbers, Sets and Set Functions. 5. Intervals. 6. n-tuples. 7. Matrices. 8. Functions. 9. Various. A List of Symbols. An Introductory Survey. I: Introduction. 1. Properties of Polynomials. 2. Elementary Inequalities. 3. Properties of Sequences. 4. Convex Functions. II: The Arithmetic, Geometric and Harmonic Means. 1. Definitions and Simple Properties. 2. The Geometric Mean-Arithmetic Mean Inequality. 3. Refinements of the Geometric Mean-Arithmetic Mean Inequality. 4. Converse Inequalities. 5. Some Miscellaneous Results. III: The Power Means. 1. Definitions and Simple Properties. 2. Sums of Powers. 3. Inequalities between the Power Means. 4. Converse Inequalities. 5. Other Means Defined Using Powers. 6. Some Other Results. IV: Quasi-Arithmetic Means. 1. Definitions and Basic Properties. 2. Comparable Means and Functions. 3. Results of Rado Popoviciu Type. 4. Further Inequalities. 5. Generalizations of the Hoelder and Minkowski Inequalities. 6. Converse Inequalities. 7. Generalizations of the Quasi-arithmetic Means. V: Symmetric Polynomial Means. 1. Elementary Symmetric Polynomials and Their Means. 2. The Fundamental Inequalities. 3.Extensions of S(r s) of Rado Popoviciu Type. 4. The Inequalities of Marcus and Lopes. 5. Complete Symmetric Polynomial Means: Whiteley Means. 6. The Muirhead Means. 7. Further Generalizations. VI: Other Topics. 1. Integral Means and Their Inequalities. 2. Two Variable Means. 3. Compounding of Means. 4. Some General Approaches to Means. 5. Mean Inequalities for Matrices. 6. Axiomatization of Means. Bibliography. Name Index. Index.

Handbook of means and their inequalities

Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules have increased in popularity, with many fruitful applications in mathematical practice. These rules require nodes with good uniform distribution properties, and digital nets and sequences in the sense of Niederreiter are known to be excellent candidates. Besides the classical theory, the book contains chapters on reproducing kernel Hilbert spaces and weighted integration, duality theory for digital nets, polynomial lattice rules, the newest constructions by Niederreiter and Xing and many more. The authors present an accessible introduction to the subject based mainly on material taught in undergraduate courses with numerous examples, exercises and illustrations.

/pdf/digital-nets-and-sequences-discrepancy-theory-and-quasi-22xlx2phr5.pdf

Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration

The Hermite-Hadamard double inequality is the first fundamental result for convex functions defined on a interval of real numbers with a natural geometrical interpretation and a loose number of applications for particular inequalities. In this monograph we present the basic facts related to Hermite- Hadamard inequalities for convex functions and a large number of results for special means which can naturally be deduced. Hermite-Hadamard type inequalities for other concepts of convexities are also given. The properties of a number of functions and functionals or sequences of functions which can be associated in order to refine the result are pointed out. Recent references that are available online are mentioned as well.

Selected Topics on Hermite-Hadamard Inequalities and Applications

Preface Organization of the Book Notations I Convex Functions and Jensen's Inequality II Some Recent Results Involving Means III Bernoulli's Inequality IV Cauchy's and Related Inequalities V Hoelder and Minkowski Inequalities VI Generalized Hoelder and Minkowski Inequalities VII Connections Between General Inequalities VIII Some Determinantal and Matrix Inequalities IX Cebysev's Inequality X Gruss' Inequality XI Steffensen's Inequality XII Abel's and Related Inequalities XIII Some Inequalities for Monotone Functions XIV Young's Inequality XV Bessel's Inequality XVI Cyclic Inequations XVII The Centroid Method in Inequalities XVII Triangle Inequalities XVIII Norm Inequalities XIX More on Norm Inequalities XX Gram's Inequality XXI Frejer-Jackson's Inequalities and Related Results XXII Mathieu's Inequality XXIII Shannon's Inequality XXIV Turan's Inequality from the Power Sum Theory XXV Continued Fractions and Pade Approximation Method XXVI Quasilinearization Methods for Proving Inequalities XXVIII Dynamic Programming and Functional Equation Approaches to Inequalities XXIX Interpolation Inequalities XXX Minimax Inequalities Name Index

https://www.springer.com/productFlyer_978-0-7923-2064-7.pdf?SGWID=0-0-1297-33677054-0

Classical and New Inequalities in Analysis

I. Landau-Kolmogorov and related inequalities.- II. An inequality ascribed to Wirtinger and related results.- III. Opial's inequality.- IV. Hardy's, Carleman's and related inequalities.- V. Hilbert's and related inequalities.- VI. Inequalities of Lyapunov and of De la Vallee Poussin.- VII. Zmorovi?'s and related inequalities.- VIII. Carlson's and related inequalities.- IX. Inequalities involving kernels.- X. Convolution, rearrangement and related inequalities.- XI. Inequalities of Caplygin type.- XII. Inequalities of Gronwall type of a single variable.- XIII. Gronwall inequalities in higher dimension.- XIV. Gronwall inequalities on other spaces: discrete, functional and abstract.- XV. Integral inequalities involving functions with bounded derivatives.- XVI. Inequalities of Bernstein-Mordell type.- XVII. Methods of proofs for integral inequalities.- XVIII. Particular inequalities.- Name Index.

Inequalities Involving Functions and Their Integrals and Derivatives

General concept of polynomials elementary inequalities zeros of polynomials special classes of polynomials extremal problems for polynomials inequalities connected with trigonometric sums.

Topics in Polynomials: Extremal Problems, Inequalities, Zeros

Preface. Basic Symbols. Basic Notations. I. Euler's phi-function. II. The arithmetical function d(n), its generalizations and its analogues. III. Sum-of-divisors function, generalizations, analogues Perfect numbers and related problems. IV. P, p, B, beta and related functions. V. omega(n), Omega(n) and related functions. VI. Function mu k-free and k-full numbers. VII. Functions pi(x), psi(x), theta(x), and the sequence of prime numbers. VIII. Primes in arithmetic progressions and other sequences. IX. Additive and diophantine problems involving primes. X. Exponential sums. XI. Character sums. XII. Binomial coefficients, consecutive integers and related problems. XIII. Estimates involving finite groups and semi-simple rings. XIV. Partitions. XV. Congruences, residues and primitive roots. XVI. Additive and multiplicative functions. Index of authors.

Dragoslav S. Mitrinović

Papers

Classical and New Inequalities in Analysis

Inequalities Involving Functions and Their Integrals and Derivatives

Topics in Polynomials: Extremal Problems, Inequalities, Zeros

Handbook of number theory

Means and Their Inequalities