On the weighted contraharmonic means

Abstract

Let $\mathcal{A}$ be a unital $C^{\ast }$-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 $\mathcal{A}$ by \begin{align*} {C}_ν(a, b):= (1-ν)ν^{-1}b + ν(1-ν)^{-1}a - \left((1-ν)a^{-1}+νb^{-1}\right)^{-1}. \end{align*} We show that \begin{align*} {C}_ν(a, b)= \displaystyle{\max_{x+y=e}}\left\{(1-ν)^{-1}\left(νa - x^*ax\right) + ν^{-1}\left((1-ν)b - y^*by\right)\right\}, \end{align*} and then apply it to present some properties of this weighted mean.