The Schwarz inequality via operator-valued inner product and the geometric operator mean

In this paper, by virtue of the Cauchy-Schwarz operator inequality due to J.I. Fujii, we show weighted mixed Schwarz operator inequalities in terms of the geometric operator mean and its Lin's type refinement. As applications, we show Wielandt type operator inequalities that refine the weighted mixed Schwarz operator inequality under some orthogonal conditions. Moreover, we show the variance-covariance operator inequality via the geometric operator mean which differs from Bhatia-Davis's one and estimate the upper bounds. By our formulation, we show a Robertson type inequality associated to a unital completely positive linear map on B(H).