Some Tensorial and Hadamard Product Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

Let $H$ be a Hilbert space. In this paper we show among others that, if $f$ is continuous differentiable convex on the open interval $I$ and $A,$ $B$ are selfadjoint operators in $B\left( H\right) $ with spectra $Sp( A) ,$ $Sp( B) \subset I,$ then we have the tensorial inequality \begin{align*} \left( f^{\prime }\left( A\right) \otimes 1\right)\left( A\otimes1-1\otimes B\right)& \geq f\left(A\right) \otimes 1-1\otimes f\left(B\right) \ & \geq \left( A\otimes 1-1\otimes B\right) \left( 1\otimes f^{\prime }\left( B\right) \right) \end{align*} and the inequality for Hadamard product \begin{align*} \left( f^{\prime }\left( A\right) A\right) \circ 1-f^{\prime }\left( A\right) \circ B& \geq \left[ f\left( A\right) -f\left( B\right) \right] \circ 1 \ & \geq A\circ f^{\prime }\left( B\right) -\left( f^{\prime }\left( B\right) B\right) \circ 1. \end{align*}.