For positive invertible operators $A$ on a Hilbert space $H$ and a fixed unit vector $x\in H$, define the normalized determinant by $\Delta_{x}(A):=\exp \left\langle \ln Ax,x\right\rangle $. In this paper we prove among others that, if $0<mI\leq A\leq MI$, then \begin{align*} 1 &eq Keft(\frac{M}{m}\right) ^{eft[ \frac{1}{2}-\frac{1}{M-m} eftangle eftěrt A-\frac{1}{2}eft( m+M\right) I\rightěrt x,x\right\rangle\right]} &eq \frac{\Delta_{x}(A)}{m^{\frac{M-eftangle Ax,x\right\rangle }{M-m}} M^{\frac{eftangle Ax,x\right\rangle -m}{M-m}}} &eq eft[ Keft( \frac{M}{m}\right) \right] ^{eft[ \frac{1}{2} + \frac{1}{M-m}eftangle eftěrt A -\frac{1}{2}eft( m+M\right) I\rightěrt x,x\right\rangle \right] }eq Keft( \frac{M}{m}\right), \end{align*} for $x\in H$, $\Vert x\Vert =1$, where $K(\cdot)$ is Kantorovich's ratio.