On generalizations of discrete and integral Cauchy-Bunyakovskii inequalities by the method of mean values. Some applications

TL;DR

This paper explores generalizations of Cauchy-Bunyakovskii inequalities using mean value methods, presents new inequalities, and discusses applications across various mathematical fields including special functions and elliptic integrals.

Contribution

It introduces new generalizations of classical inequalities and demonstrates their applications to special functions and other mathematical areas.

Findings

Derived a new inequality involving maximum and minimum values.

Provided estimates for Euler gamma and incomplete gamma functions.

Outlined potential further generalizations including q-integrals.

Abstract

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.