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Cauchy--Schwarz-type inequalities for additive functions

Zsolt Páles, Mahmood Kamil Shihab

Abstract

The main goal of this paper is to show that if a real valued function defined on a groupoid satisfies a certain Levi--Civita-type functional equation, then it also fulfills a Cauchy--Schwarz-type functional inequality. In particular, if the groupoid is the multiplicative structure of commutative ring, then we can establish the existence of nontrivial additive functions satisfying inequalities connected to the multiplicative structure.

Cauchy--Schwarz-type inequalities for additive functions

Abstract

The main goal of this paper is to show that if a real valued function defined on a groupoid satisfies a certain Levi--Civita-type functional equation, then it also fulfills a Cauchy--Schwarz-type functional inequality. In particular, if the groupoid is the multiplicative structure of commutative ring, then we can establish the existence of nontrivial additive functions satisfying inequalities connected to the multiplicative structure.
Paper Structure (4 sections, 13 theorems, 75 equations)

This paper contains 4 sections, 13 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. The inequality $A(x*y)^2\leq A(x*x)A(y*y)$
  3. The inequality $A(x*x)A(y*y)\leq A(x*y)^2$
  4. Consequences of systems of Levi--Civita-type functional equations

Key Result

Theorem 1

Let $(G,*)$ be a groupoid and let $A:G\to\mathbb{R}$ be a function. Assume that there exist $n\in\mathbb{N}$ and functions $f_1,\dots,f_n:G\to\mathbb{R}$ such that $A$ the Levi--Civita-type functional equation holds for all $x,y\in G$. Then, $A$ satisfies the functional inequality for all $x,y\in G$.

Theorems & Definitions (28)

  • Theorem
  • proof
  • Theorem
  • proof
  • Theorem
  • proof
  • Theorem
  • proof
  • Corollary
  • proof
  • ...and 18 more