On the weighted contraharmonic means

Let A be a unital C*-algebra with unit e and let ? ? [0, 1]. We introduce the concept of the ?-weighted contraharmonic mean of two positive definite elements a and b of A by C?(a, b) := 2 ((1 ? ?)a + ?b)?((1 ? ?)a?1 + ?b?1)?1. When ? ? (0, 1), we show that C?(a, b) = max x+y=e {2(1 ? ?)a ? (1 ? ?)?1x*ax + 2?b ? ??1y*by}, and then apply it to present some properties of this weighted mean.