Finite Field Tarski-Maligranda Inequalities

TL;DR

This paper extends Tarski-Maligranda inequalities to finite fields, providing new bounds for norms in sub-normed linear spaces over such fields.

Contribution

It introduces finite field analogues of classical inequalities, expanding their applicability to algebraic structures over finite fields.

Findings

Derived inequalities for norms over finite fields.

Established bounds that generalize classical Tarski-Maligranda inequalities.

Demonstrated the inequalities in the context of sub-normed linear spaces.

Abstract

Let $\mathbb{F}$ be a sub-modulus field such that $2 \neq 0$. Let $\mathcal{X}$ be a sub-normed linear space over $\mathbb{F}$. Then we show that \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|x-y\|, \|y-x\|\}-(\|x\|+\|y\|) \end{align*} and \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \|x\|+\|y\|-\frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|y-x\|, \|x-y\|\}. \end{align*} Above inequalities are finite field versions of important Tarski-Maligranda inequalities obained by Maligranda [\textit{Banach J. Math. Anal., 2008}].