Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.05194

Non-Archimedean Cauchy-Schwarz Angle-Length and Chebyshev Arithmetic Mean Inequalities

K. Mahesh Krishna
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $\mathbb{K}$ be a non-Archimedean valued field. Let $n \in \mathbb{N}$. For every $(a_j)_{j=1}^n, (b_j)_{j=1}^n \in \mathbb{K}^n$, we show that \begin{align*} \left|\sum_{j=1}^{n}a_jb_j\right|^2\leq \max\left\{\left|\sum_{j=1}^n a_j^2\right|\left|\sum_{k=1}^nb^2_k\right|, \max_{1\leq j<k \leq n}|a_jb_k-a_kb_j|^2\right\}, \end{align*} \begin{align*} \left|\sum_{j=1}^n a_j^2\right|\left|\sum_{k=1}^nb^2_k\right|\leq \max\left\{\left|\sum_{j=1}^n a_jb_j\right|^2, \max_{1\leq j<k \leq n}|a_jb_k-a_kb_j|^2\right\}, \end{align*} \begin{align*} \left|AM(a_j)_{j=1}^n\right|\left|AM(b_j)_{j=1}^n\right|\leq \max\left\{ \left|AM(a_jb_j)_{j=1}^n\right|, \frac{1}{|n|^2}\max_{1\leq j < k \leq n}|a_j-a_k||b_j-b_k|\right\}, \end{align*} \begin{align*} \left|AM(a_jb_j)_{j=1}^n\right|\leq \max\left\{ \left|AM(a_j)_{j=1}^n\right|\left|AM(b_j)_{j=1}^n\right|, \frac{1}{|n|^2}\max_{1\leq j < k \leq n}|a_j-a_k||b_j-b_k|\right\}, \end{align*} where $AM$ denotes the arithmetic mean. First and second are non-Archimedean versions of Cauchy-Schwarz angle-length and third and fourth are Chebyshev arithmetic mean inequalities. Unlike in the Archimedean case, no conditions are required to derive non-Archimedean Chebyshev arithmetic mean inequalities.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

K. Mahesh Krishna. 2026-08-03. Non-Archimedean Cauchy-Schwarz Angle-Length and Chebyshev Arithmetic Mean Inequalities. https://arxiv.org/abs/2608.05194

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗