Some Additive Inequalities for Heinz Operator Mean

In this paper we obtain some new additive inequalities for Heinz operator mean, namely the operator $H_{\nu }\left( A,B\right) :=\frac{1}{2}\left( A\sharp _{\nu }B+A\sharp _{1-\nu }B\right) $ where $A\sharp _{\nu }B:=A^{1/2}\left( A^{-1/2}BA^{-1/2}\right) ^{\nu }A^{1/2}$ is the weighted geometric mean for the positive invertible operators $A$ and $B,$ and $\nu \in \left[ 0,1\right] .$

Keywords:

Youngs Inequality, Real functions Arithmetic mean-Geometric mean inequality,

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