The upper bound of a reserve Hölder's type operator inequality and its applications. (Q1870926)

Let \(A\) and \(B\) be two commuting positive operators on a Hilbert space \(H\) such that \[ 0<m_{1}\leq A\leq M_{1}\quad \text{and}\quad 0<m_{2}\leq A\leq M_{2}\quad(m_{1} <M_{1},\;m_{2}<M_{2}). \] The main result of this paper is the upper estimate \[ \langle A^{p}x,x\rangle^{1/p}\langle B^{q}x,x\rangle^{1/q}-\lambda\langle ABx,x\rangle\leq M_{1}M_{2}F_{0}(\lambda,m_{1}/M_{1},m_{2}/M_{2},p), \] valid for all unit vectors \(x\in H\) and all real numbers \(\lambda>0\) and \(p>1\). Here \(1/p+1/q=1.\) This extends a number of well-known inequalities (such as the Greub-Rheinboldt inequality) and also a result due to \textit{S. A. Gheorghiu} [ Bull. Math. Soc. Roum. Sci. 35, 117--119 (1933; Zbl 0008.34504)]. As a consequence, it is shown the existence of a similar upper bound for \[ \langle A^{p}x,x\rangle^{1/p}\langle B^{q}x,x\rangle^{1/q}-\lambda\langle B^{q}\natural_{1/p}A^{p}x,x\rangle, \] where the (positive and invertible) operators \(A\) and \(B\) are not necessarily commuting; \(B^{q}\natural_{1/p}A^{p}\) represents the Kubo-Ando \(1/p\)-geometric mean of \(B^{q}\) and \(A^{p}\). It is shown that \(C\natural_{s}D=C^{1/2}\) \((C^{-1/2}DC^{-1/2})^{s}C^{1/2}\), cf. \textit{F. Kubo} and \textit{T. Ando} [ Math. Ann. 246, 205--224 (1980; Zbl 0412.47013)]. Note: There is an obvious misprint in the title: ``reserve'' should be replaced by ``reverse''.