Upper estimations on integral operator means (Q2839860)

Let \(m\) be a symmetric operator mean in the sense of Kubo-Ando and \(A,B\) be positive operators on a Hilbert space. The path \(m_t\) between \(A\) and \(B\) is defined by the inductive relation NEWLINE\[NEWLINE Am_{(2k+1)/2^{n+1}}B=(Am_{k/2^{n}}B)m(Am_{(k+1)/2^{n}}B)= (Am_{(k+1)/2^{n}}B)m(Am_{k/2^{n}}B) NEWLINE\]NEWLINE with initial conditions NEWLINE\[NEWLINE Am_0B=A,~~Am_{1/2}B=AmB,~~Am_1B=B. NEWLINE\]NEWLINE If \(m_t\) satisfies NEWLINE\[NEWLINE (Am_rB)m_t(Am_sB)=Am_{(1-t)r+ts}B NEWLINE\]NEWLINE for all \(r,s,t\in [0,1]\), then the integral mean for \(m\) is defined by NEWLINE\[NEWLINE A\widetilde{m}B=\int_0^1Am_tB\, dt. NEWLINE\]NEWLINE The authors prove that NEWLINE\[NEWLINE A\widetilde{m}B\leq \frac{sA+(1-s)B+Am_sB}{2} NEWLINE\]NEWLINE for all \(s\in [0,1]\). Some similar inequalities are also discussed.