On the lines of Fuglede-Kadison's determinant and ours, we 
define an operator valued one for positive invertible operators 
on a Hilbert space: For a unital positive linear map $\Phi$, put 
$\Delta_\Phi(A)=\exp\Phi(\log A)$. Then we show a parametrized estimation 
$\Phi(A^{t})^{1/t}\ge\Delta_\Phi(A)\equiv\lim_{t\to0}\Phi(A^{t})^{1/t}
\ge\Phi(A^{-t})^{-1/t}$. Based on this, we show a reverse Oppenheim 
inequality and Ando's product formula for Hadamard products.