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

Let K be a non-Archimedean valued field. Let n ∈ N. For every (aⱼ)ⱼ₌₁ⁿ, (bⱼ)ⱼ₌₁ⁿ ∈ Kⁿ, we show that {align*} |∑ⱼ₌₁ⁿa_jb_j|^2≤ max\{|∑ⱼ₌₁^n a_j^2||∑ₖ₌₁^nb^2_k|, max1≤ j<k ≤ n|a_jb_k-a_kb_j|^2\}, {align*} {align*} |∑ⱼ₌₁^n a_j^2||∑ₖ₌₁^nb^2_k|≤ max\{|∑ⱼ₌₁^n a_jb_j|^2, max1≤ j<k ≤ n|a_jb_k-a_kb_j|^2\}, {align*} {align*} |AM(a_j)ⱼ₌₁^n||AM(b_j)ⱼ₌₁^n|≤ max\{ |AM(a_jb_j)ⱼ₌₁^n|, 1/|n|^2max1≤ j < k ≤ n|a_j-a_k||b_j-b_k|\}, {align*} {align*} |AM(a_jb_j)ⱼ₌₁^n|≤ max\{ |AM(a_j)ⱼ₌₁^n||AM(b_j)ⱼ₌₁^n|, 1/|n|^2max1≤ j < k ≤ n|a_j-a_k||b_j-b_k|\}, {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.