P. S. Bullen

Means and their inequalities

The theory of means has its roots in the work of the Pythagoreans who introduced the harmonic, geometric, and arithmetic means with reference to their theories of music and arithmetic. Later, Pappus introduced seven other means and gave the well-known elegant geometric proof of the celebrated inequalities among the harmonic, geometric, and arithmetic means. Nowadays, the families and types of means that are being investigated by researchers and the variety of questions that are being asked about them are beyond the scope of any single survey, with the voluminous book Handbook of Means and Their Inequalities by P. S. Bullen being the best such reference in this direction. The theory of means has grown to occupy a prominent place in mathematics with hundreds of papers on the subject appearing every year. The strong relations and interactions of the theory of means with the theories of inequalities, functional equations, and probability and statistics add greatly to its importance.

The Burkill approximately continuous integral

This paper defines descriptive, Riemann, and constructive integrals equivalent to the approximately continuous integral of Burkill. 1980 Mathematics subject classification (Amer. Math. Soc.): 26 A 39.

The Quasi-Arithmetic Means

The power means n [r] (a;w), reR, defined in the previous chapter can be looked at in the following way; for each reR define a function φ as follows: Φ(x) = xr, r ≠ 0, Φ(x) = log x, r = 0, then

Symmetric Polynomial Means

M_n^{[r]}(\underline a ;\underline w ) = {\phi ^{ - 1}}\quad (\frac{1}{{{w_n}}}\sum\limits_{i = 1}^n {{w_i}\;\phi ({a_i})} ).

Further Means, Axiomatics and Other Topics

(1) This suggests the following definitions.

Handbook of means and their inequalities

The power means are defined using the convex, or concave, power, logarithmic and exponential functions. In this chapter means are defined using arbitrary convex and concave functions by a natural extension of the classical definitions and analogues of the basic results of the earlier chapters are investigated. First however we take up the problem of different convex functions defining the same means; the case of equivalent means. The generalizations (GA) and (r;s), their converses and the Rado-Popoviciu type extensions are studied under the topic of comparable means. The definition can be further extended although this leads to the topics of functional equations and functional inequalities so is not followed in detail.

Dictionary of Inequalities

The elementary and complete symmetric polynomials have a history that goes back to Newton at the beginning of the modern mathematical era. They are used to define means that generalize the geometric and arithmetic means in a completely different way to the generalizations of Chapters III and IV. These new means give extensions of the geometric mean-arithmetic mean inequality. In this chapter we study the properties of these means. In addition generalizations of these means due to Whiteley and Muirhead are discussed

The p n -integral

The theory of means has its roots in the work of the Pythagoreans who introduced the harmonic, geometric, and arithmetic means with reference to their theories of music and arithmetic. Later, Pappus introduced seven other means and gave the well-known elegant geometric proof of the celebrated inequalities among the harmonic, geometric, and arithmetic means. Nowadays, the families and types of means that are being investigated by researchers and the variety of questions that are being asked about them are beyond the scope of any single survey, with the voluminous book Handbook of Means and Their Inequalities by P. S. Bullen being the best such reference in this direction. The theory of means has grown to occupy a prominent place in mathematics with hundreds of papers on the subject appearing every year. The strong relations and interactions of the theory of means with the theories of inequalities, functional equations, and probability and statistics add greatly to its importance.