Some properties of h-MN-convexity and Jensen's type inequalities

In this work, we introduce the class of $h$-${\rm{MN}}$-convex functions by generalizing the concept of ${\rm{MN}}$-convexity and combining it with $h$-convexity. Namely, Let $I,J$ be two intervals subset of $\left(0,\infty\right)$ such that $\left(0,1\right)\subseteq J$ and $\left[a,b\right]\subseteq I$. Consider a non-negative function $h: (0,\infty)\to \left(0,\infty\right)$ and let ${\rm{M}}:\left[0,1\right]\to \left[a,b\right]$ $(0<a<b)$ be a Mean function given by ${\rm{{\rm{M}}}}\left(t\right)={\rm{{\rm{M}}}}\left( {h(t);a,b} \right)$; where by ${\rm{{\rm{M}}}}\left( {h(t);a,b} \right)$ we mean one of the following functions: $A_h\left( {a,b} \right):=h\left( {1 - t} \right)a + h(t) b$, $G_h\left( {a,b} \right)=a^{h(1-t)} b^{h(t)}$ and $H_h\left( {a,b} \right):=\frac{ab}{h(t) a + h\left( {1 - t} \right)b} = \frac{1}{A_h\left( {\frac{1}{a},\frac{1}{b}} \right)}$; with the property that ${\rm{{\rm{M}}}}\left( {h(0);a,b} \right)=a$ and ${\rm{M}}\left( {h(1);a,b} \right)=b$. A function $f : I \to \left(0,\infty\right)$ is said to be $h$-${\rm{{\rm{MN}}}}$-convex (concave) if the inequality \begin{align*} f \left({\rm{M}}\left(t;x, y\right)\right) \le (\ge) \, {\rm{N}}\left(h(t);f (x), f (y)\right), \end{align*} holds for all $x,y \in I$ and $t\in [0,1]$, where M and N are two mean functions. In this way, nine classes of $h$-${\rm{MN}}$-convex functions are established and some of their analytic properties are explored and investigated. Characterizations of each type are given. Various Jensen's type inequalities and their converses are proved.