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.
In many practical
investigations, it is necessary to bound one quantity by another. The
classical inequalities are very useful for this purpose.
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
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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