The contraharmonic mean of two positive semidefinite Hermitian matrices A and B is defined by the relation where 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 It is shown that
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.