Inequalities for mixed means

In this paper, we prove the inequality $\mathcal{M}_p\circ\mathcal{M}_q(x)\leq\mathcal{M}_q\circ\mathcal{M}_p(x)$ between mixed power means for absolutely continuous measures (with respect to the Lebesgue measure) and obtain several more inequalities for mixed means. For all integral inequalities, their discrete analogues are discussed and it is explored when equality holds.