• 1  A. Kufner et L.E. Persson, Weighted Inequalities of Hardy Type, NJ, World Scientific Publishing Co (...)
• 2  A Kufner, L. Maligranda et L.E. Persson, The Hardy Inequality. About its History and Some Related (...)

where f is a measurable and non-negative function on (0, ∞), p > 1, α < p − 1. First I present Hardy’s original motivation from around 1915 and some historical facts until Hardy in 1925 finally proved the first version of (1) (with α = 0). After that I present the easiest proof of (1) I know, which was not discovered by Hardy himself. I continue by presenting some steps in the remarkable development of (1) to what today is called Hardy type inequalities, see e.g. the books1 and the references therein. In particular, I present a choice of examples of interesting questions and results from each of the Chapters in the book2 including the following:

• 3  C. Niculescu et L.E. Persson, Convex Functions and their Applications - A Contemporary Approach. C (...)

5Convexity is one of the most fundamental concepts in analysis and various applications (see e.g. the eight books listed in Appendix I, in particular3 [B] below).

• 4  L.E. Persson et N. Samko, Classical and New Inequalities via Convexity and Interpolation. à paraît (...)

7I begin by pointing out the fact that several (more than 10) of the most common inequalities in analysis books are more or less simple consequences of the concept of convexity. After that I shortly describe some examples of the interplay between the developments of (complex and real) interpolation theory and the theory and understanding of other inequalities. The aim is to have a more unified approach to derive inequalities than in other inequality books, see4.

• 5  L. Larson, L. Maligranda, J. Pecaric et L.E. Persson, Multiplicative Inequalities of Carlson Type (...)

8Just as one example how it is possible to develop these ideas further I present some generalizations of Carlson’s inequalities, which were important for the development of the real interpolation method (J.L. Lions, J. Peetre, and others). After that I present some further developments, which help to understand and further develop the Peetre ± interpolation method and its applications (e.g. with this method we can even interpolate between certain Orlicz spaces), see also5.