A family of inequalities originating from coding of messages

A nice generalization of the first inequality is proved: Let ∗ be one of the four operations +, ×, min and max on an appropriate interval J of R. Let a, b ∈ Jn. Denote by a ∗ a the n × n matrix ai,j = ai ∗ aj. Then the matrix a ∗ a is more different from b ∗ b than a ∗ b is from b ∗ a. Precisely, if ∣A∣=∑1⩽i,j⩽n∣ai,j∣, then ∥a ∗ a − b ∗ b∥ ⩾ ∥a ∗ b − b ∗ a∥.