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The contraharmonic mean of HSD matrices. (English) Zbl 0641.15009

The contraharmonic mean of two positive semidefinite Hermitian matrices A and B is defined by the relation $C\left(A,B\right)=A+B-2\left(A:B\right),$ where $A:B=A{\left(A+B\right)}^{-1}B$ is the so-called parallel addition introduced by W. N. Anderson jun. and R. J. Duffin [J. Math. Anal. Appl. 26, 576-594 (1969; Zbl 0177.049)]. The dual of the contraharmonic mean of A and B is given by ${C}^{\text{'}}\left(A,B\right)=C{\left({A}^{-1},{B}^{-1}\right)}^{-1}·$ It is shown that

${C}^{\text{'}}\left(A,B\right)=A:B+2\left(A:B\right)C{\left(A,B\right)}^{-1}\left(A:B\right)=\left(A{\left(A:B\right)}^{-1}A\right):\left(B{\left(A:B\right)}^{-1}B\right)·$

With the aid of the contraharmonic mean and its dual the authors study fixed point problems, the monotonicity behaviour of C(A,B), an infinite family of means for positive semidefinite Hermitian matrices that generalize C(A,B), inverse mean problems, and connections between C(A,B) and least square problems.

Reviewer: A.R.Kräuter

##### MSC:
 15A45 Miscellaneous inequalities involving matrices 15A24 Matrix equations and identities 15A27 Commutativity of matrices