Mantra Patel
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http://doi.org/10.37648/ijrst .v16i04.003
For four positive-semidefinite matrices of rank at most one, the average norm of a product of four independently sampled factors is at least the average over permutations of the four factors. The assertion holds in every positive finite dimension, for every unitarily invariant norm, and with arbitrary nonnegative eigenvalue scales. A rank-one reduction leads to a scalar polynomial on a six-dimensional cube. Its tensor-product Bernstein coefficients are certified by nonnegative coefficients or exact weighted arithmetic-geometric mean inequalities. A supplied integer certificate and two separately implemented Python checkers cover all 4096 Bernstein indices. The result is restricted to four rank-one factors and does not resolve the conjecture for general positive-semidefinite matrices.
Keywords: positive-semidefinite matrices; rank-one matrices; unitarily invariant norms; Bernstein basis; arithmetic-geometric mean inequality
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