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Non-Archimedean Tarski-Maligranda Inequalities

K. Mahesh Krishna

Abstract

In 1930, Tarski observed that \begin{align*} \bigg||r|-|s|\bigg|=|r-s|+ |r+s|-(|r|+|s|), \quad \forall r, s \in \mathbb{R}. \end{align*} In 2008, Maligranda converted the previous equality into inequalities that are valid in every normed linear space. We derive non-Archimedean versions of Tarski-Maligranda inequalities. Difference between Archimedean and non-Archimedean inequalities is surprising.

Non-Archimedean Tarski-Maligranda Inequalities

Abstract

In 1930, Tarski observed that \begin{align*} \bigg||r|-|s|\bigg|=|r-s|+ |r+s|-(|r|+|s|), \quad \forall r, s \in \mathbb{R}. \end{align*} In 2008, Maligranda converted the previous equality into inequalities that are valid in every normed linear space. We derive non-Archimedean versions of Tarski-Maligranda inequalities. Difference between Archimedean and non-Archimedean inequalities is surprising.
Paper Structure (3 sections, 2 theorems, 21 equations)

This paper contains 3 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Non-Archimedean Tarski-Maligranda Inequalities
  3. Conclusions

Key Result

Theorem 1.1

MALIGRANDA (Tarski-Maligranda Inequalities) Let $\mathcal{X}$ be a NLS. Then for all $x, y \in \mathcal{X}\setminus\{0\}$, and

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 2.1
  • proof