  In this preprint we consider generalizations of discrete and integral
Cauchy--Bunyakovskii inequalities by the method of mean values with some
applications. Mostly the material is compiled as a short survey but some
results are proved. Main results are listed, including an interesting
inequality with maximum and minimum values. Some applications are considered
from different fields of mathematics. Among them are estimates for some special
functions, including Euler gamma and incomplete gamma function, the Legendre
complete elliptic integrals of the first kind. Also some further possible
generalizations are considered and outlined, including generalizations of the
Acz\'el and Minkovskii inequalities, a case of spaces with sign-indefinite
form, the Jackson's $q$-integrals.
