$\mathfrak{H}_p\mathfrak{H}_q$-convex functions and generalization of the Hölder, Minkowski, and Muirhead inequalities

Let $\mathfrak{M}$, $\mathfrak{N}$ be any means. Let $\mathfrak{H}_p$ be a power mean with exponent $p$. A function $f$ is called $\mathfrak{MN}$-convex if for any $x$ and $y$ from the domain of $f$ the inequality$f(\mathfrak{M}(x,y))\leqslant\mathfrak{N}(f(x),f(y))$ holds. In this paper the method of constructing $\mathfrak{H}_p\mathfrak{H}_q$-convex functions is proposed. For such functions generalizations of Cauchy–Schwarz, Hölder, Minkowski, Mahler, and Muirhead inequalities are obtained.