On some matrix mean inequalities via the log-convexity property

Abstract

Let f : [0, 1] ?? [0,+?) be a log-convex function, 0 ? ? ? 1/2 ? ? ? 1 and m be a positive integer. Then by using the Jensen?s inequality we prove that ?m/? (fm(0)?? f m(1) ? f m(?))+ rm (f(0)m/2 ? f m/2(?))2 ? (f(0)?? f (1))m ? f m(?) and (1??)m/1?? (fm(0)?? fm(1) ? fm(?))+ r?m(f(1)m/2 ? fm/2(?))2 ? ( f (0)?? f (1))m ? f m(?). Here, ?? denotes the weighted arithmetic mean, and rm, r?m are two positive constants. Moreover, by choosing suitable log-convex functions, we derive new refinements of several classical inequalities that relate the difference between the arithmetic-power, arithmetic-harmonic, and arithmetic-geometric means for both scalars and matrices and matrices, as well as matrix norms and determinants.