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

Abstract

In this work, we introduce the class of $h$-$\mathrm{MN}$-convex functions by generalizing the concept of $\mathrm{MN}$-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 $\mathrm{M}:\left[0,1\right]\to \left[a,b\right]$ $(0<a<b)$ be a Mean function given by $\mathrm{M}\left(t\right)=\mathrm{M}\left(h(t);a,b\right)$; where by $\mathrm{M}\left(h(t);a,b\right)$ 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 $\mathrm{M}\left(h(0);a,b\right)=a$ and $\mathrm{M}\left(h(1);a,b\right)=b$. A function $f:I\to \left(0,\infty \right)$ is said to be $h$-$\mathrm{MN}$-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 $h$-$\mathrm{MN}$-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.

arXiv comment27 pages. Journal of Interdisciplinary Mathematics 2019

References in corpus (2)

• Generalized Convexity and Inequalities
• Definitions of h-logaritmic, h-geometric and h-multi convex functions and some inequalities related to them

Cited by in corpus (1)

• Popoviciu's type inequalityies for h-MN-convex functions