A general class of inequalities with mixed means

Suppose $(T,\Sigma,\mu)$ is a space with positive measure, $f\colon\mathbb R\to\mathbb R$ is a strictly monotone continuous function, and $\mathfrak G(T)$ is the set of real $\mu$-measurable functions on $T$. Let $x(\cdot)\in\mathfrak G(T)$ and $(f\circ x)(\cdot)\in L_1(T,\mu)$. Comparison theorems are proved for the means $\mathfrak M_{(T,\mu,f)}\bigl (x(\cdot)\bigr)$ and the mixed means $\mathfrak M_{(T_1,\mu _1,f_1)}\bigl(\mathfrak M_{(T_2,\mu_2,f_2)}\bigl(x(\cdot)\bigr)\bigr)$ these inequalities imply analogs and generalizations of some classical inequalities, namely those of Hölder, Minkowski, Bellman, Pearson, Godunova and Levin, Steffensen, Marshall and Olkin, and others. These results are a continuation of the author's studies.