Let K be a non-Archimedean valued field. Let n∈N. For every (aj)j=1n,(bj)j=1n∈Kn
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, we show that align* |∑_j=1ⁿa_jb_j|²≤ {|∑_j=1ⁿ a_j²||∑_k=1ⁿb²_k|, ₁≤ j<k ≤ n|a_jb_k-a_kb_j|²}, align* align* |∑_j=1ⁿ a_j²||∑_k=1ⁿb²_k|≤ {|∑_j=1ⁿ a_jb_j|², ₁≤ j<k ≤ n|a_jb_k-a_kb_j|²}, align* align* |AM(a_j)_j=1ⁿ||AM(b_j)_j=1ⁿ|≤ { |AM(a_jb_j)_j=1ⁿ|, (1/|n|²)₁≤ j < k ≤ n|a_j-a_k||b_j-b_k|}, align* align* |AM(a_jb_j)_j=1ⁿ|≤ { |AM(a_j)_j=1ⁿ||AM(b_j)_j=1ⁿ|, (1/|n|²)₁≤ 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.