On the weighted contraharmonic means

Let A be a unital C^*-algebra with unit e and let ν∈(0, 1). We introduce the concept of the ν-weighted contraharmonic of two positive definite elements a and b of A by {align*} {C}ν(a, b):= (1-ν)ν⁻¹b + ν (1-ν)⁻¹a - ((1-ν)a⁻¹+ν b⁻¹)⁻¹. {align*} We show that {align*} {C}ν(a, b)= {maxx+y=e}\{(1-ν)⁻¹(ν a - x^*ax) + ν⁻¹((1-ν)b - y^*by)\}, {align*} and then apply it to present some properties of this weighted mean.