2020/01/27 by Kian, Mohsen, Alomari, Mohammad W.
#15A18 #15A42 #15A45 #47A56 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2001.10013
We show that if f is a non-negative superquadratic function, then A\mapstoTrf(A) is a superquadratic function on the matrix algebra. In particular, \tr f( (A + B)/(2) ) +\tr f(| (A - B)/(2)|) ≤ \frac\tr f( A ) + \tr f( B ) 2 holds for all positive matrices A,B. In addition, we present a Klein's inequality for superquadratic functions as Tr[f(A)-f(B)-(A-B)f'(B)]≥ Tr[f(|A-B|)] for all positive matrices A,B. It gives in particular an improvement of the Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.