Why Mathematical Inequalities?

Richard Bellman stated succinctly, at the Second International Conference on Mathematical Inequalities, Oberwolfach, Germany, July 30-August 5, 1978, " ... there are three reasons for the study of inequalities: practical, theoretical, and aesthetic.

1. In many practical investigations, it is necessary to bound one quantity by another. The classical inequalities are very useful for this purpose.
2. From the theoretical point of view, very simple questions give rise to entire theories. For example, we may ask when the non-negativity of one quantity implies that of another. This simple question leads to the theory of positive operators and the theory of differential inequalities, etc.
3. Finally, let us turn to the aesthetic aspects. As has been pointed out, beauty is in the eyes of the beholder. However, it is generally agreed that certain pieces of music, art or mathematics are beautiful. There is an elegance to inequalities that make them very attractive ..."

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