[Paper Review] Some properties of h-MN-convexity and Jensen's type inequalities

Mohammad W. Alomari|arXiv (Cornell University)|Oct 10, 2017

Mathematical Inequalities and Applications30 references3 citations

TL;DR

This paper introduces the novel class of $h$-MN-convex functions by unifying $h$-convexity with mean-type inequalities, generalizing classical convexity and existing $h$-convex classes. It establishes nine distinct types of $h$-MN-convexity, derives characterizations, and proves new Jensen-type inequalities and their converses under various conditions on $h$, including submultiplicativity and supermultiplicativity.

ABSTRACT

In this work, we introduce the class of $h$-${ m{MN}}$-convex functions by generalizing the concept of ${ m{MN}}$-convexity and combining it with $h$-convexity. Namely, Let $I,J$ be two intervals subset of $\left(0,\infty ight)$ such that $\left(0,1 ight)\subseteq J$ and $\left[a,b ight]\subseteq I$. Consider a non-negative function $h: (0,\infty) o \left(0,\infty ight)$ and let ${ m{M}}:\left[0,1 ight] o \left[a,b ight] $ $(0

Motivation & Objective

• To generalize classical convexity by combining $h$-convexity with mean functions (M and N) to define a new class of $h$-MN-convex functions.
• To explore the analytic properties and characterizations of nine distinct types of $h$-MN-convex functions based on different mean functions (arithmetic, geometric, harmonic).
• To establish new Jensen-type inequalities and their converses for $h$-MN-convex functions under specific conditions on the weight function $h$.
• To unify and extend existing classes such as $s$-convex, Godunova-Levin, and $P$-functions within the $h$-MN-convex framework.
• To provide a systematic framework for inequalities involving weighted means and $h$-functions, with applications to product and harmonic mean forms.

Proposed method

• Define $h$-MN-convexity via the inequality $ f(M(t;x,y)) \leq N(h(t); f(x), f(y)) $, where $M$ and $N$ are mean functions and $h$ is a non-negative function on $(0,\infty)$.
• Construct three base mean functions: $A_h(a,b) = h(1-t)a + h(t)b$, $G_h(a,b) = a^{h(1-t)}b^{h(t)}$, and $H_h(a,b) = \frac{ab}{h(t)a + h(1-t)b}$, forming nine combinations of $M$ and $N$.
• Use the $h$-cord concept to geometrically interpret $h$-convexity, generalizing the standard chord to $L(t;h)$, with $h(t)=t$ recovering the classical chord.
• Derive characterizations of $h$-MN-convexity by analyzing the behavior of $f$ at interior points using mean-based interpolation and functional inequalities.
• Prove weighted Jensen-type inequalities for $h$-MN-convex functions using finite sequences and weight functions $w_k/W_n$, with results depending on $h$'s submultiplicativity or supermultiplicativity.
• Establish reversed inequalities for $h$-MN-concave functions and analyze cases where $h(t) = t$, $h(t) = 1$, $h(t) = 1/t$, and $h(t) = t^s$ to recover known convexity classes.

Experimental results

Research questions

• RQ1 How can $h$-convexity be generalized by incorporating mean functions $M$ and $N$ to form a unified class of $h$-MN-convex functions?
• RQ2 What are the nine distinct types of $h$-MN-convexity formed by combining arithmetic, geometric, and harmonic means with $h$-weights?
• RQ3 What characterizations and structural properties define each type of $h$-MN-convex function?
• RQ4 What new Jensen-type inequalities emerge for $h$-MN-convex functions, and how do they depend on the multiplicative behavior of $h$?
• RQ5 In what way do known convexity classes (e.g., $s$-convex, Godunova-Levin, $P$-functions) arise as special cases of $h$-MN-convexity?

Key findings

• Nine distinct classes of $h$-MN-convex functions are established by combining three mean functions ($A_h$, $G_h$, $H_h$) with $h$-weighting, each with unique characterizations.
• For $h$-MN-convex functions, the inequality $ f(M(t;x,y)) \leq N(h(t); f(x), f(y)) $ holds for all $x,y \in I$ and $t \in [0,1]$, generalizing classical convexity.
• When $h$ is submultiplicative and $f$ is $H_tG_h$-convex, the product inequality $ \prod_{k=1}^n [f(x_k)]^{h(w_k/W_n)} \leq \prod_{k=1}^n \left\{ f(m)^{h(\cdot)} f(M)^{h(\cdot)} \right\} $ holds.
• For $h$-MN-convex functions with $h$ submultiplicative and $f$ positive $H_tH_h$-convex, the harmonic-type inequality $ \left(\sum \frac{h(w_k/W_n)}{f(x_k)} \right)^{-1} \leq \left(\sum \frac{h(\cdot) f(m) + h(\cdot) f(M)}{f(m)f(M)} h(w_k/W_n) \right)^{-1} $ is derived.
• The results recover known classes: $h(t)=t$ gives classical convexity, $h(t)=1$ gives $P$-functions, $h(t)=1/t$ gives Godunova-Levin, and $h(t)=t^s$ gives $s$-convexity.
• The framework allows extension of Theorem 22 and Corollary 23 from Varošanec (2007) to the $h$-MN-convex setting, with reversed inequalities under concavity assumptions.