A. M. Fink

Bessel’s Inequality

Let us consider the problem of the best approximation of a vector x by vectors of an orthonormal system from a Hilbert space X. For every system of numbers λ1,...,λ2 we have

Shannon’s Inequality

||x - \sum\limits_{k = 1}^n {{\lambda _k}{x_k}|{|^2}} \geqslant 0.

Some Recent Results Involving Means

The concept of the centroid, introduced most likely by Archimedes, can be applied in solving various Mathematical problems. We mention, for example, the papers of K. F. Gauss [1] and L. Fejer [2]. Here we shall give a chronological account of the use of the centroid in developing inequalities, pointing to some priorities which are neglected in the literature.

Fejér-Jackson’s Inequalities and Related Results

The notion of the Shannon entropy appears frequently and is important in many works. In this Chapter we will review some of the characterizations of it and of the concept of the gain of information with functional inequalities. Similarly, we shall present a characterization of Renyi’s generalized concept of information measure and gain of information with the aid of functional inequalities. These inequalities, to be discussed, have also other interpretations.

Convex Minimax Inequalities-Equalities

Let x 1,...,x n be vectors of a unitary space X. Then

Dynamic Programming and Functional Equation Approaches to Inequalities

G\,\left( {{x_1}\,,\,...\,,{x_n}} \right)\, = \,\left[{\begin{array}{*{20}{c}}{\left( {{x_1},\,{x_n}} \right)\,...\,\left( {{x_1}\,,\,{x_n}} \right)} \\ \vdots \\ {\left( {{x_1}\,,\,{x_n}} \right)\,...\,\left( {{x_n}\,,{x_n}} \right)} \end{array}} \right]