Further extension of the Heinz-Kato-Furuta inequality

Let $T$ be a bounded operator on a Hilbert space $\mathfrak{H},$ and $A,B$ positive definite operators. Kato has shown that if $\Vert T x \Vert \leq \Vert A x \Vert$ and $\Vert T^{*} y \Vert \leq \Vert B y \Vert$ for all $x, y \in \mathfrak{H}$, then $|(Tx , y)| \leq ||f(A)x|| \; ||g(B)y||,$ where $f(t), g(t)$ are operator monotone functions defined on $[0, \infty )$ such that $f(t) g(t)=t$. Furuta has shown that $|(T|T|^{\alpha +\beta -1}x,y)|\leq ||A^{\alpha }x|| ||B^{\beta }y||, \text{ where } 0 \leq \alpha ,\beta \leq 1, 1\leq \alpha + \beta .$ Let $f(t), g(t)$ be any continuous operator monotone functions, and set $h(t) = f(t) g(t) /t$ for $t >0.$ We will show that $Th(|T|)$ is well defined and $|(T h(|T|)x,y)| \leq ||f(A)x|| \; ||g(B)y||.$ Moreover, we will extend this result for unbounded closed operators densely defined on $\mathfrak{H}.$