Matrix Fejér and Levin-Stečkin Inequalities

Abstract: Fe{j}\'er and Levin-Ste\v{c}kin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fej\'{e}r type for both convex and log-convex functions. For matrix Levin-Ste\v{c}kin type, we present more rigorous results involving the partial Loewner ordering for Hermitian matrices.