On the weighted contraharmonic means

Abstract

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 mean of two positive
definite elements $a$ and $b$ of $\mathscr{A}$ by
\begin{align*}
{C}_{\nu}(a, b):= 2\big((1-\nu)a+\nu b\big) - \left((1-\nu)a^{-1}+\nu b^{-1}\right)^{-1}.
\end{align*}
When $\nu\in(0, 1)$, we show that
\begin{align*}
{C}_{\nu}(a, b)= \displaystyle{\max_{x+y=e}}\left\{2(1-\nu)a - (1-\nu)^{-1}x^*ax + 2\nu b - \nu^{-1}y^*by\right\},
\end{align*}
and then apply it to present some properties of this weighted mean.