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\fancyhead[LO]{\textbf{Non-Archimedean Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality}}
\fancyhead[RE]{\textbf{K. Mahesh Krishna}}


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{\bf\large{{Non-Archimedean Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality}}}\\
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\textbf{K. Mahesh Krishna}\\
School of Mathematics and Natural Sciences\\
Chanakya University Global Campus\\
NH-648, Haraluru Village\\
Devanahalli Taluk, 	Bengaluru  North District\\
Karnataka State, 562 110, India\\
Email: kmaheshak@gmail.com\\

Date: \today
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\textbf{Abstract}: In 1992, Hudzik and Landes  \textit{[Math. Ann.]} derived a breakthrough generalization of the triangle inequality for two nonzero elements in normed linear spaces, which was generalized to finitely many nonzero elements independently in 2006 by Dragomir \textit{[Bull. Aust. Math. Soc.]} and in 2007 by Kato, Saito and Tamura \textit{[Math. Inequal. Appl]}. We derive a non-Archimedean version of Hudzik-Landes-Dragomir-Kato-Saito-Tamura inequality.\\
\textbf{Keywords}:  Normed linear space, Triangle inequality, Non-Archimedean linear space, Ultra-norm.\\
\textbf{Mathematics Subject Classification (2020)}:  12J25, 46S10.\\

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\section{Introduction}

Let $\mathcal{X}$ be a normed linear space (NLS). From the definition of the norm, we have the triangle inequality 
\begin{align}\label{TI}
\|x+y\|\leq \|x\|+\|y\|, \quad \forall x, y \in \mathcal{X}.
\end{align}
In 1992, Hudzik and Landes derived a breakthrough generalization of Inequality (\ref{TI}) which is valid for any two nonzero elements in a NLS \cite{HUDZIKLANDES}.
\begin{theorem} \cite{HUDZIKLANDES} \label{HKT} (\textbf{Hudzik-Landes Inequlaity})
Let $\mathcal{X}$ be a NLS. Then for all $x, y \in \mathcal{X}\setminus\{0\}$, 
\begin{align}\label{HL}
\|x+y\|\leq \|x\|+\|y\|-\left(2-\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\|\right)\min\{\|x\|, \|y\|\}.
\end{align}
\end{theorem}
We note that, in 2006, Maligranda independently derived Inequality (\ref{HL}) \cite{MALIGRANDA}.
It is natural to ask for a generalization of Inequality (\ref{HL}) to more than two non-zero vectors. This is done independently by Dragomir in 2006 \cite{DRAGOMIR} and by Kato, Saito and Tamura in 2007 \cite{KATOSAITOTAMURA}.
\begin{theorem} \cite{KATOSAITOTAMURA, DRAGOMIR} \label{HLKSTT}
(\textbf{Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality})
Let $\mathcal{X}$ be a NLS and $n \in \mathbb{N}$. Then for all $x_1, \dots, x_n \in \mathcal{X}\setminus\{0\}$, 	we have 
\begin{align*}
\left\|\sum_{j=1}^{n}x_j\right\|\leq \sum_{j=1}^{n}\|x_j\|-\left(n-\left\|\sum_{j=1}^{n}\frac{x_j}{\|x_j\|}\right\|\right)\min_{1\leq k \leq n} \|x_k\|.
\end{align*}	
\end{theorem}
It is natural and important to ask what are non-Archimedean versions of Theorems \ref{HKT} and \ref{HLKSTT}? We answer the question by deriving non-Archimedean version of Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality (Theorem \ref{NAHLDKSTT}).





\section{Non-Archimedean Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality}
Let $\mathbb{K}$ be a field. Recall that a map $|\cdot|: \mathbb{K} \to [0, \infty)$ is said to be a non-Archimedean  valuation  if following conditions holds.
\begin{enumerate}[\upshape(i)]
	\item If $\lambda  \in \mathbb{K}$ is such that $|\lambda|=0$, then $\lambda=0$.
	\item $|\lambda \mu|=|\lambda||\mu|$ for all $\lambda, \mu  \in \mathbb{K}$.
	\item (Ultra-triangle inequality) $|\lambda+\mu|\leq \max\{|\lambda|, |\mu|\}$ for all $\lambda, \mu \in \mathbb{K}$.
\end{enumerate}
In this case, $\mathbb{K}$ is called as non-Archimedean valued field \cite{SCHIKHOF}.  Let $\mathcal{X}$ be a vector space over a non-Archimedean valued field $\mathbb{K}$ with valuation $|\cdot|$. Recall that a map $\|\cdot\|: \mathcal{X} \to [0, \infty)$ is said to be a non-Archimedean norm if following conditions holds.
\begin{enumerate}[\upshape(i)]
	\item If $x \in \mathcal{X}$ is such that $\|x\|=0$, then $x=0$.
	\item $\|\lambda x\|=|\lambda|\|x\|$ for all $\lambda \in \mathbb{K}$, for all $x \in \mathcal{X}$.
	\item (Ultra-norm inequality) $\|x+y\|\leq \max\{\|x\|, \|y\|\}$ for all $x,y \in \mathcal{X}$.
\end{enumerate}
In this case, $\mathcal{X}$ is called as non-Archimedean linear space (NALS) \cite{GARCIASCHIKHOF}. We first derive non-Archimedean version of Inequality (\ref{HL}).
\begin{theorem} (\textbf{Non-Archimedean Hudzik-Landes Inequality})
Let $\mathcal{X}$ be a NALS over $\mathbb{K}$. Then for all $x, y \in \mathcal{X}\setminus\{0\}$ with $\|x\|, \|y\| \in \mathbb{K}$ it holds
\begin{align*}
	\|x+y\|&\leq \min \left\{\big|\|x\|\big|\max\left\{\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\|, \left|\frac{1}{\|x\|}-\frac{1}{\|y\|}\right|\|y\|\right\}, \big|\|y\|\big|\max\left\{\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\|, \left|\frac{1}{\|x\|}-\frac{1}{\|y\|}\right|\|x\|\right\}\right\}.
\end{align*}
\end{theorem}
\begin{proof}
	Let $x, y \in \mathcal{X}\setminus\{0\}$ with $\|x\|, \|y\| \in \mathbb{K}$. Then 
	\begin{align*}
		\|x+y\|&=\left\|\|x\|\left(\frac{x}{\|x\|}+\frac{y}{\|y\|}\right)+\left(1-\frac{\|x\|}{\|y\|}\right)y\right\|\\
			&\leq \max\left\{\left\|\|x\|\left(\frac{x}{\|x\|}+\frac{y}{\|y\|}\right)\right\|, \left\|\left(1-\frac{\|x\|}{\|y\|}\right)y\right\|\right\}\\
		&= \max\left\{\big|\|x\|\big|\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\|, \big|\|x\|\big|\left|\frac{1}{\|x\|}-\frac{1}{\|y\|}\right|\|y\|\right\}
	\end{align*}
and 
	\begin{align*}
	\|x+y\|&=\left\|\left(1-\frac{\|y\|}{\|x\|}\right)x+\|y\|\left(\frac{x}{\|x\|}+\frac{y}{\|y\|}\right)\right\|\\
	&\leq \max\left\{\left\|\left(1-\frac{\|y\|}{\|x\|}\right)x\right\|, \left\|\|y\|\left(\frac{x}{\|x\|}+\frac{y}{\|y\|}\right)\right\|\right\}\\
	&\leq \max\left\{\big|\|y\|\big|\left|\frac{1}{\|y\|}-\frac{1}{\|x\|}\right|\|x\|, \big|\|y\|\big|\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\| \right\}.
\end{align*}
Therefore 
\begin{align}\label{P1}
\|x+y\|\leq	\big|\|x\|\big|\max\left\{\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\|, \left|\frac{1}{\|x\|}-\frac{1}{\|y\|}\right|\|y\|\right\}
\end{align}
and 
\begin{align}\label{P2}
\|x+y\|\leq  \big|\|y\|\big|\max\left\{\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\|, \left|\frac{1}{\|x\|}-\frac{1}{\|y\|}\right|\|x\|\right\}
.
\end{align}
Inequalities (\ref{P1})and  (\ref{P2}) give 
\begin{align*}
	\|x+y\|\leq \min \left\{\big|\|x\|\big|\max\left\{\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\|, \left|\frac{1}{\|x\|}-\frac{1}{\|y\|}\right|\|y\|\right\}, \big|\|y\|\big|\max\left\{\left\|\frac{x}{\|x\|}+\frac{y}{\|y\|}\right\|, \left|\frac{1}{\|x\|}-\frac{1}{\|y\|}\right|\|x\|\right\}\right\}.
\end{align*}
\end{proof}
Note the additional assumption $\|x\|, \|y\| \in \mathbb{K}$ in the previous theorem. The reason is that, since the norm is a real number, we generally do not have a guarantee that it belongs to the given non-Archimedean field. Now we derive non-Archimedean version of  Theorem \ref{HLKSTT}.
\begin{theorem} \label{NAHLDKSTT} (\textbf{Non-Archimedean Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality})
Let $\mathcal{X}$ be a NALS over $\mathbb{K}$ and $n \in \mathbb{N}$. Then for all $x_1, \dots, x_n \in \mathcal{X}\setminus\{0\}$ with $\|x_1\|, \dots, \|x_n\| \in \mathbb{K}$ it holds
\begin{align*}
\left\|\sum_{j=1}^{n}x_j\right\|\leq\min_{1\leq k \leq n}\left\{\big|\|x_k\|\big|\max\left\{\left\|\sum_{j=1}^{n}\frac{x_j}{\|x_j\|}\right\|, \max_{1\leq j \leq n}\left|\frac{1}{\|x_j\|}-\frac{1}{\|x_k\|}\right|\|x_j\|\right\}\right\}.
\end{align*}
\end{theorem}
\begin{proof}
Let $x_1, \dots, x_n \in \mathcal{X}\setminus\{0\}$ with $\|x_1\|, \dots, \|x_n\| \in \mathbb{K}$. Let $1\leq k \leq n$ be fixed. Then 
\begin{align*}
	\left\|\sum_{j=1}^{n}x_j\right\|&=\left\|\sum_{j=1}^{n}\frac{\|x_k\|x_j}{\|x_j\|}+\sum_{j=1}^{n}\left(1-\frac{\|x_k\|}{\|x_j\|}\right)x_j\right\|\\
	&\leq \max\left\{\left\|\sum_{j=1}^{n}\frac{\|x_k\|x_j}{\|x_j\|}\right\|, \left\|\sum_{j=1}^{n}\left(1-\frac{\|x_k\|}{\|x_j\|}\right)x_j\right\|\right\}\\
	&=\max\left\{\big|\|x_k\|\big|\left\|\sum_{j=1}^{n}\frac{x_j}{\|x_j\|}\right\|, \big|\|x_k\|\big| \left\|\sum_{j=1}^{n}\left(\frac{1}{\|x_k\|}-\frac{1}{\|x_j\|}\right)x_j\right\|\right\}\\
	&=\big|\|x_k\|\big|\max\left\{\left\|\sum_{j=1}^{n}\frac{x_j}{\|x_j\|}\right\|,  \left\|\sum_{j=1}^{n}\left(\frac{1}{\|x_k\|}-\frac{1}{\|x_j\|}\right)x_j\right\|\right\}\\
		&\leq \big|\|x_k\|\big|\max\left\{\left\|\sum_{j=1}^{n}\frac{x_j}{\|x_j\|}\right\|,  \max_{1\leq j \leq n}\left\|\left(\frac{1}{\|x_k\|}-\frac{1}{\|x_j\|}\right)x_j\right\|\right\}\\
			&=\big|\|x_k\|\big|\max\left\{\left\|\sum_{j=1}^{n}\frac{x_j}{\|x_j\|}\right\|,  \max_{1\leq j \leq n}\left|\frac{1}{\|x_k\|}-\frac{1}{\|x_j\|}\right|\|x_j\|\right\}.
\end{align*}
By varying $k$ and taking minimum in the right side of previous inequality gives 
	\begin{align*}
		\left\|\sum_{j=1}^{n}x_j\right\|\leq\min_{1\leq k \leq n}\left\{\big|\|x_k\|\big|\max\left\{\left\|\sum_{j=1}^{n}\frac{x_j}{\|x_j\|}\right\|, \max_{1\leq j \leq n}\left|\frac{1}{\|x_j\|}-\frac{1}{\|x_k\|}\right|\|x_j\|\right\}\right\}.
	\end{align*}
\end{proof}
Now we derive continuous version of Theorem \ref{NAHLDKSTT}. 
\begin{theorem}
Let $\mathcal{X}$ be a NALS over $\mathbb{K}$ and $(\Omega, \mu)$ be a non-Archimedean measure space. Let $\phi: \Omega\to \mathcal{X}\setminus\{0\}$ be a measurable function such that $\|\phi(\alpha)\|\in \mathbb{K}$ for every $\alpha \in \Omega$. Then 
\begin{align*}
	\left\|\int_{\Omega}\phi(\alpha)\,d\mu(\alpha)\right\|&\leq\inf_{\beta \in \Omega}\left\{\big|\|\phi(\beta)\|\big|\max\left\{\left\|\int_{\Omega}\frac{\phi(\alpha)}{\|\phi(\alpha)\|}\,d\mu(\alpha)\right\|, \sup_{\alpha \in \Omega}\left|\frac{1}{\|\phi(\alpha)\|}-\frac{1}{\|\phi(\beta)\|}\right|\|\phi(\alpha)\|\right\}\right\}.
\end{align*}	
\end{theorem}
\begin{proof}
Let $\phi: \Omega\to \mathcal{X}\setminus\{0\}$ be a measurable function such that $\|\phi(\alpha)\|\in \mathbb{K}$ for every $\alpha \in \Omega$. 	Let $\beta \in \Omega$ be fixed. Then 
		\begin{align*}
		\left\|\int_{\Omega}\phi(\alpha)\,d\mu(\alpha)\right\|&=\left\|\int_{\Omega}\frac{\|\phi(\beta)\|\phi(\alpha)}{\|\phi(\alpha)\|}\,d\mu(\alpha)+\int_{\Omega}\left(1-\frac{\|\phi(\beta)\|}{\|\phi(\alpha)\|}\right)\phi(\alpha)\,d\mu(\alpha)\right\|\\
		&\leq \max\left\{\left\|\int_{\Omega}\frac{\|\phi(\beta)\|\phi(\alpha)}{\|\phi(\alpha)\|}\,d\mu(\alpha)\right\|, \left\|\int_{\Omega}\left(1-\frac{\|\phi(\beta)\|}{\|\phi(\alpha)\|}\right)\phi(\alpha)\,d\mu(\alpha)\right\|\right\}\\
		&=\max\left\{\big|\|\phi(\beta)\|\big|\left\|\int_{\Omega}\frac{\phi(\alpha)}{\|\phi(\alpha)\|}\,d\mu(\alpha)\right\|, \big|\|\phi(\beta)\|\big| \left\|\int_{\Omega}\left(\frac{1}{\|\phi(\beta)\|}-\frac{1}{\|\phi(\alpha)\|}\right)\phi(\alpha)\,d\mu(\alpha)\right\|\right\}\\
		&=\big|\|\phi(\beta)\|\big|\max\left\{\left\|\int_{\Omega}\frac{\phi(\alpha)}{\|\phi(\alpha)\|}\,d\mu(\alpha)\right\|,  \left\|\int_{\Omega}\left(\frac{1}{\|\phi(\beta)\|}-\frac{1}{\|\phi(\alpha)\|}\right)\phi(\alpha)\,d\mu(\alpha)\right\|\right\}\\
		&\leq \big|\|\phi(\beta)\|\big|\max\left\{\left\|\int_{\Omega}\frac{\phi(\alpha)}{\|\phi(\alpha)\|}\,d\mu(\alpha)\right\|,  \sup_{\alpha \in \Omega}\left\|\left(\frac{1}{\|\phi(\beta)\|}-\frac{1}{\|\phi(\alpha)\|}\right)\phi(\alpha)\right\|\right\}\\
		&= \big|\|\phi(\beta)\|\big|\max\left\{\left\|\int_{\Omega}\frac{\phi(\alpha)}{\|\phi(\alpha)\|}\,d\mu(\alpha)\right\|,  \sup_{\alpha \in \Omega}\left|\frac{1}{\|\phi(\beta)\|}-\frac{1}{\|\phi(\alpha)\|}\right|\|\phi(\alpha)\|\right\}.
\end{align*}
By varying $\beta$ and taking infimum in the previous inequality gives 
\begin{align*}
	\left\|\int_{\Omega}\phi(\alpha)\,d\mu(\alpha)\right\|\leq\inf_{\beta \in \Omega}\left\{\big|\|\phi(\beta)\|\big|\max\left\{\left\|\int_{\Omega}\frac{\phi(\alpha)}{\|\phi(\alpha)\|}\,d\mu(\alpha)\right\|, \sup_{\alpha \in \Omega}\left|\frac{1}{\|\phi(\alpha)\|}-\frac{1}{\|\phi(\beta)\|}\right|\|\phi(\alpha)\|\right\}\right\}.
\end{align*}	
\end{proof}

\section{Conclusions}
\begin{enumerate}
	\item In 1992, Hudzik and Landes improved centuries old triangle inequality in normed linear spaces \cite{HUDZIKLANDES}.
	\item In 2006, Dragomir extended Hudzik-Landes inequality for more than two vectors \cite{DRAGOMIR}.
	\item In 2007, Kato, Saito and Tamura extended Hudzik-Landes inequality without knowing the work of Dragomir \cite{KATOSAITOTAMURA}.
	\item In this article, we extended centuries old ultra-norm inequality.  
\end{enumerate}




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\end{thebibliography}

 


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