An arithmetic-geometric-harmonic mean inequality involving Hadamard products

AbstractGiven matrices of the same size, A = [aij] and B = [bij], we define their Hadamard product to be A ∘ B = [aijbij]. We show that if xi > 0 and q ⩾ p ⩾ 0, then the n × n matrices xixjxi+xj,,xi−1+xj−1xixj,and xip+xjpxiq+xjq are positive definite, and we relate these facts to some matrix valued arithmetic- geometric-harmonic mean inequalities—some of which involve Hadamard products and others unitarily invariant norms. It is known that if A is positive semidefinite then max{|A∘B|:|B|⩽1} = max aij, where |·| denotes the spectral norm. We show that the converse of this statement is false and give a useful partial converse