Finite Field Tarski-Maligranda Inequalities

Let $\mathbb{F}$ be a sub-modulus field such that $2\mathrm{\ne }0$. 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 [$\mathit{\text{Banach}}\mathit{\text{J.}}\mathit{\text{Math.}}\mathit{\text{Anal.,}}\mathit{\text{2008}}$].