Résumé
Different types of inequalities are very important in various areas of mathematics and its applications. Today the knowledge about inequalities has been developed to be an independent area with many papers, Journals, conferences and books (29 of them are listed in Appendix I).
Interpolation theory was partly motivated by developing a theory for better understanding of some inequalities (e.g. Hausdorff-Young’s inequality) and has now been developed to a fairly independent area of great interest also for several applications (see e.g. the eight books listed in Appendix I).
Convexity is one of the most fundamental concepts in analysis and various applications (see e.g. the eight books listed in Appendix I, in particular [B] below).
The core of this lecture is to give a number of examples how the developments of these areas have supported each other and that it is still an intensive interplay between the continued developments.
I 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, see [C].
Just 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 also [A].
A number of open questions are raised.
