Superquadratic functions and eigenvalue inequalities

Abstract

There exist two major subclasses in the class of superquadratic functions, one contains concave and decreasing functions and the other, contains convex and monotone increasing functions. We use this fact and present eigenvalue inequalities in each case. The nature of these functions enables us to apply our results in two directions. First improving some known results concerning eigenvalues for convex functions and second, obtaining some complimentary inequalities for some other functions. We will give some examples to support our results as well. As applications, some subadditivity inequalities for matrix power functions have been presented. In particular, if $X$ and $Y$ are positive matrices, then $$(X+Y)^q \leq U^*X^qU+ V^*Y^qV + 2 \left(\lambda_1(X+Y)-\lambda_n(X+Y)\right)^q\qquad (q\in[1,2])$$ for some unitaries $U,V\in\mathbb{M}_n$