Inequalities between the sum of powers and the exponential of sum of positive and commuting selfadjoint operators. (Q2897396)

Motivated by a paper of \textit{B. Belaïdi}, \textit{A. El Farissi} and \textit{Z. Latreuch} [``Inequalities between the sum of power and the exponential of sum of nonnegative sequence'', RGMIA Research Report Collection 11, No.~1 (2008), \url{http://rgmia.org/papers/v11n1/Belaidi-Report.pdf}], where the inequality NEWLINE\[NEWLINE \frac {e^p}{p^p}\sum _{i=1}^n x_i^p\leq \exp \left \{ \sum _{i=1}^n x_i\right \} NEWLINE\]NEWLINE was proved for real nonnegative numbers \(x_1,\dots ,x_n\) and \(p\geq 1\), the authors establish a similar inequality for operators in a Hilbert space. The main result of the paper reads as follows.NEWLINENEWLINETheorem: Let \(A_1,\dots ,A_n\) be self-adjoint positive operators in a Hilbert space and \(p\geq 1\) be an integer. Then the operator NEWLINE\[NEWLINE B:=\exp \left \{\sum _{i=1}^n A_i\right \}- \frac {e^p}{p^p}\sum _{i=1}^n A_i^p NEWLINE\]NEWLINE is positive. The constant \(\frac {e^p}{p^p}\) in the above expression is the best possible.