An arithmetic-harmonic-mean inequality for nonnegative definite matrices (Q1375097)

The parallel sum \(A:B\) of two nonnegative definite \(m \times m\) matrices \(A\) and \(B\) is defined to the nonnegative definite matrix \(A(A + B)^{+}B\) where \((A + B)^{+}\) is the Moore-Penrose inverse of \(A + B\). The author proves the ``arithmetic-harmonic mean inequality'' \(\alpha _{1}A_{1} + \ldots + \alpha _{n}A_{n} \geq (\alpha ^{-1}_{1}A_{1}):\ldots :(\alpha ^{-1}_{n}A_{n})\) where the \(A_{i}\) are nonnegative definite \(m \times m\) matrices, the weights \(\alpha _{i}\) are positive real numbers summing to 1, and \(\geq \) denotes the Loewner partial ordering on the set of real symmetric \(m \times m\) matrices (\(A \geq B\) if and only if \(A-B\) is nonnegative definite). This inequality was proved earlier in the special case where all the weights \(\alpha _{i} = 1/n\) by \textit{W. N. Anderson}, jun. and \textit{R. J. Duffin} [J. Math. Anal. Appl. 26, 576-594 (1969; Zbl 0177.04904)].