Let $\mathscr{ A}$ be a unital $C^*$-algebra with unit $e$ and let $\nu\in(0, 1)$. We introduce the concept of the $\nu$-weighted contraharmonic of two positive definite elements $a$ and $b$ of $\mathscr{ A}$ by \begin{ align*} { C}_{ \nu}(a, b):= (1-\nu)\nu^{ -1}b + \nu (1-\nu)^{ -1}a - \left((1-\nu)a^{ -1}+\nu b^{ -1}\right)^{ -1}. \end{ align*} We show that \begin{ align*} { C}_{ \nu}(a, b)= \displaystyle{ \max_{ x+y=e}}\left\{ (1-\nu)^{ -1}\left(\nu a - x^*ax\right) + \nu^{ -1}\left((1-\nu)b - y^*by\right)\right\}, \end{ align*} and then apply it to present some properties of this weighted mean.
Categories: math.FA math.OA