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On the Generalizations of the Cauchy-Schwarz-Bunyakovsky Inequality with Applications to Elasticity

Dimitra Labropoulou, Thanasis Labropoulos, Panayiotis Vafeas, Dimitris M. Manias

Abstract

In this article we present both the discrete and the integral form of Cauchy-Bunyakovsky-Schwarz (CBS) inequality, some important generalizations in the n-dimensional Euclidean space and in linear subspaces of it, as well as the strengthened CBS. The last CBS inequality plays an important role in elasticity problems. A geometrical interpretation and a collection of the most important proofs of it are, also, presented.

On the Generalizations of the Cauchy-Schwarz-Bunyakovsky Inequality with Applications to Elasticity

Abstract

In this article we present both the discrete and the integral form of Cauchy-Bunyakovsky-Schwarz (CBS) inequality, some important generalizations in the n-dimensional Euclidean space and in linear subspaces of it, as well as the strengthened CBS. The last CBS inequality plays an important role in elasticity problems. A geometrical interpretation and a collection of the most important proofs of it are, also, presented.
Paper Structure (8 sections, 12 theorems, 114 equations)

This paper contains 8 sections, 12 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Cauchy-Bunyakovsky-Schwarz Inequality
  3. Cauchy Inequality
  4. Cauchy-Bunyakovsky-Schwarz Inequality
  5. Geometrical Interpretation of the $\rm CBS$ Inequality
  6. Triangular Inequality
  7. A Cauchy-Bunyakovsky-Schwarz Type Inequality
  8. Strengthened Cauchy-Bunyakovsky-Schwarz Inequality for Elasticity

Key Result

Proposition 1

If $\alpha _1 ,\,\alpha _2 ,\,...,\alpha _n$ are non-negative real numbers, then If in addition the numbers $\alpha _1 ,\,\alpha _2 ,\,...,\alpha _n$ are different from zero, then Equality is true only when $\alpha _1 = \alpha _2 = ... = \alpha _\nu$.

Theorems & Definitions (12)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 2
  • Proposition 7
  • Theorem 3
  • ...and 2 more