A Class of Norm Inequalities for Operator Monotone Functions and Hyponormal Operators

Let \(\Psi \) $\mathrm{\Psi }$ , \(\Phi \) $\mathrm{\Phi }$ be s.n. functions, \(p\ge 2,\) $p\ge 2,$ and let \(\varphi \) $\phi$ be an operator monotone function on \([0,\infty )\) $[0,\infty )$ such that \(\varphi (0)=0.\) $\phi (0)=0.$ If are such that A and B are strictly accretive and then also and \(\begin{aligned}{} & {} \vert {\;\!\!\vert {AX\varphi {(B)}-\varphi {(A)}XB}\vert \;\!\!}\vert _\Psi \\{} & {} \qquad \le \left\| \,\!\sqrt{\varphi \Bigl ({\tfrac{A+A^*}{2}}\Bigr )-\tfrac{A+A^*}{2}\varphi ^\prime \Bigl ({\tfrac{A+A^*}{2}}\Bigr )} \Bigl ({\tfrac{A+A^*}{2}}\Bigr )^{-1}\!A(AX-XB)B\Bigl ({\tfrac{B+B^*}{2}}\Bigr )^{-1}\!\!\right. \\{} & {} \qquad \quad \left. \sqrt{\varphi \Bigl ({\tfrac{B+B^*}{2}}\Bigr )-\tfrac{B+B^*}{2}\varphi ^\prime \Bigl ({\tfrac{B+B^*}{2}}\Bigr )}\,\right\| _\Psi \,\!\!. \end{aligned}\) $$\begin{array}{cc}\hfill & {||AX\phi (B)-\phi (A)XB||}_{\mathrm{\Psi }}\hfill \\ \hfill & \le \left.∥,,\sqrt{\phi (\frac{A+A^{\ast }}{2})-\frac{A+A^{\ast }}{2}{\phi }^{\prime }(\frac{A+A^{\ast }}{2})},(,\frac{A+A^{\ast }}{2},{)}^{-1},,A,(AX-XB),B,(,\frac{B+B^{\ast }}{2},{)}^{-1},,\right)\hfill \\ \hfill & {\left(\sqrt{\phi (\frac{B+B^{\ast }}{2})-\frac{B+B^{\ast }}{2}{\phi }^{\prime }(\frac{B+B^{\ast }}{2})},\right.∥}_{\mathrm{\Psi }}.\hfill \end{array}$$ under any of the following conditions: (a)

Both A and B are normal,