$\mathfrak{K}_{\varphi}\mathfrak{K}_{\psi}$-convex functions and generalizations of classical inequalities

A function $f$ is called $\mathfrak{MN}$-convex, if for any $x$ and $y$ from the domain of $f$ inequality $f(\mathfrak{M}(x,y))\leqslant\mathfrak{N}(f(x),f(y))$ holds, where $\mathfrak{M}$ and $\mathfrak{N}$ are means. In this paper geometric interpretation of $\mathfrak{MN}$-convexity of a function is obtained, where $\mathfrak{M}$ and $\mathfrak{N}$ are Kolmogorov's means. For such functions analogies of rearrangement, Popovicu's, Chebyshev's sum and Hermite–Hadamar's inequalities are obtained.