Inequalities for the normalized determinant of positive operators in Hilbert spaces via Tominaga and Furuichi results

For positive invertible operators \(A\) on a Hilbert space \(H\) and a fixed unit vector \(x∈ H,\) define the normalized determinant by \(Δₓ(A):=exp ln Ax,x\). In this paper, we prove among others that, if \(0<mI≤ A≤ MI,\) then {aligned}1&≤ exp ln S( ( M/m) 1/2 I-1/M-m A-1/2( m+M) I) x,x \\ &≤ {Δ ₓ(A)}{mM- Ax,x /M-mMAx,x -m/M-m}≤ S( M/m ){aligned}for \(x∈ H,\) \( x =1,\) where \(S( · )\) is Specht's ratio.