THE INEQUALITIES OF CAUCHY-BUNIAKOVSKY AND HELDER AND THEIR GENERALIZATIONS

Abstract

Formulation of the problem. Various mathematical literature is devoted to classical inequalities. The Cauchy-Buniakowski and Helder inequalities are at the heart of the geometry of unitary and normed spaces. The article considers the generalization of these constructions - polylinear forms and inequalities for them. The content of generalized inequalities consists in estimating a polylinear form from a system of vectors through their norms. The form itself in appearance is a generalization of the scalar product of an arbitrary number of vectors. It is essential that the proofs are carried out by elementary methods, without using means of mathematical analysis. It is known that the Cauchy-Buniakowski inequality is a partial case of Helder's inequality. The paper shows that, on the contrary, the second of these inequalities can be derived from the first. The application of proven inequalities to specific vectors yields well-known results, in particular, inequalities for power averages and some others that the authors did not encounter in the mathematical literature. Inequalities can be transferred to vectors from infinite-dimensional sequence spaces. They can also be used to find the extremum of some functions and in preparation for the Olympics.

Materials and methods. To prove the generalized Cauchy-Buniakovsky inequality, the Cauchy inequality was used for non-negative numbers, which are the coordinates of vectors in a multidimensional space. With a certain choice of such vectors, the generalized Helder inequality is proved from this inequality. By choosing vectors in various ways, you can get different meaningful inequalities.

The results. The generalized Cauchy-Buniakovsky, Helder inequalities, the inequality for power averages, and some others are proven.

Conclusions. The application of the generalized Cauchy-Buniakovsky and Helder inequalities to various systems of vectors with non-negative coordinates gives inequalities - both well-known and new and quite meaningful. Their proof is reduced only to the selection of the desired system of vectors. In this way, it is possible to easily prove inequalities that can be found at mathematical Olympiads.