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.

Pi and the AGM

Handbook of means and their inequalities

There are many possible ways of introducing the number π, associated with the circle. We shall obtain π from the complex exponential function 

$$ \exp \,z = 1 + \frac{z} {{2!}} + \cdots . $$

.

What is π

We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log(2) or π on a modest work station in a few hours run time.

/pdf/on-the-rapid-computation-of-various-polylogarithmic-2jkwpfausz.pdf

On the rapid computation of various polylogarithmic constants

We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is $a_n+1 := a_n + 2b_n / 3$ and b_n+1 := [formula cannot be replicated]. The limit of this iteration is identified in terms of the hypergeometric function ₂F₁ (1/3, 2/3; 1 ; ·), which supports a particularly simple cubic transformation.

/pdf/a-cubic-counterpart-of-jacobi-s-identity-and-the-agm-1g6as1mb89.pdf

A cubic counterpart of Jacobi’s identity and the AGM

In geometric function theory, generalized elliptic integrals and functions arise ?from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to π. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them.

/pdf/generalized-elliptic-integrals-and-modular-equations-mqmdgncsyi.pdf

Pi and the AGM

Citations

Handbook of means and their inequalities

What is π

On the rapid computation of various polylogarithmic constants

A cubic counterpart of Jacobi’s identity and the AGM

Generalized elliptic integrals and modular equations

Related Papers (5)

Ramanujan’s Notebooks: Part V

A Course of Modern Analysis

Ramanujan's Notebooks

Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity

The lost notebook and other unpublished papers