Cauchy-Bunyakovsky inequalities: brief historical information, generalizations, and applications

The modern theory of inequalities is based on a relatively small number of classical results. Moreover, each of the main famous inequalities, as it exists, is overgrown with numerous refinements, generalizations, and applications that contribute to a deeper understanding of its essence and expanding the set of possible applications. The purpose of this survey on the Cauchy-Bunyakovsky inequalities (discrete and integral) is to present brief historical information about these inequalities, their generalizations, and refinements, including a number of little-known for a wide circle of mathematicians. For example, Milne's inequalities or the Carlitz-Daykin-Eliezer theorem, the Gini and Rado mean. We will also present some of the author's results on the refinement of the Cauchy - Bunyakovsky inequalities. These results are based on the use of the averages theory. Some applications are considered: analogs of the Cauchy-Bunyakovsky inequality and their refinements for the Jackson q-integral and Lorentz spaces, the uncertainty principle for the discrete Fourier transform, exact estimates of Legendre elliptic integrals, probabilistic estimates of the correlation and regression coefficients, and a number of others. *) The entrance is the same