[Paper Review] On some inequalities for different kinds of convexity

Merve Avcı Ardıç, M. Emіn Özdemіr|arXiv (Cornell University)|May 17, 2012

Mathematical Inequalities and Applications4 references3 citations

TL;DR

This paper establishes new inequalities for various generalized convex functions, including $φ_s$-convex, $φ$-Godunova-Levin, $φ$-$P$, and $φ$-quasi-convex functions, by analyzing the composition $f \circ \varphi$. It proves that under specific conditions on $\varphi$ (linearity or convexity) and $f$ (monotonicity), the composition inherits generalized convexity properties, and introduces a new class of $φ$-quasi-convex functions with associated integral and weighted inequalities.

ABSTRACT

In this paper, we obtained some inequalities for ϕ_{s}-convex function, ϕ-Godunova-Levin function, ϕ-P-function and log-ϕ-convex function. Finally, we defined the class of ϕ-quasi-convex functions and we examined some properties of this class.

Motivation & Objective

• To extend classical convexity inequalities to generalized classes of functions involving a transformation $\varphi$.
• To investigate the behavior of the composition $f \circ \varphi$ when $f$ belongs to specific generalized convexity classes.
• To define and analyze a new class of functions: $\varphi$-quasi-convex functions.
• To derive new integral and weighted inequalities for these generalized convex functions.
• To establish sufficient conditions under which $f \circ \varphi$ inherits convexity properties from $f$ and $\varphi$.

Proposed method

• Define generalized convexity classes via a function $\varphi$ and a control function $h$, including $\varphi_h$-convexity.
• Introduce $\varphi$-quasi-convexity as a new class where $f(t\varphi(x) + (1-t)\varphi(y)) \leq \max\{f(\varphi(x)), f(\varphi(y))\}$.
• Use induction and recursive substitution to prove weighted inequalities for $\varphi_s$-convex and $\varphi$-Godunova-Levin functions.
• Apply monotonicity and convexity assumptions on $f$ and $\varphi$ to derive composition properties.
• Use variable substitution and integration over $[0,1]$ to transform inequalities into integral forms.
• Leverage the structure of convex combinations and normalization of weights ($\sum t_i = 1$) to generalize results.

Experimental results

Research questions

• RQ1 Under what conditions does the composition $f \circ \varphi$ preserve $s$-convexity when $f$ is $\varphi_s$-convex?
• RQ2 How does the $\varphi$-Godunova-Levin property of $f$ affect the behavior of $f \circ \varphi$ when $\varphi$ is linear or convex?
• RQ3 What are the implications of $\varphi$-quasi-convexity for the integral mean of $f$ over $[\varphi(x), \varphi(y)]$?
• RQ4 Can weighted inequalities be established for $\varphi_s$-convex and $\varphi$-Godunova-Levin functions using recursive decomposition?
• RQ5 How does the monotonicity of $f$ interact with the convexity of $\varphi$ to preserve generalized convexity in the composition?

Key findings

• If $f$ is $\varphi_s$-convex and $\varphi$ is linear, then $f \circ \varphi$ is $s$-convex in the second sense.
• If $f$ is $\varphi_s$-convex and increasing, and $\varphi$ is convex, then $f \circ \varphi$ is $s$-convex in the second sense.
• For $\varphi_s$-convex $f$ and weights $t_i$ summing to 1, $f\left(\sum t_i \varphi(x_i)\right) \leq \sum t_i^s f(\varphi(x_i))$.
• If $f$ is $\varphi$-Godunova-Levin and $\varphi$ is linear, then $f \circ \varphi$ belongs to the class $Q(I)$ of quasi-convex functions.
• For $\varphi$-quasi-convex $f$, the inequality $\frac{1}{\varphi(y)-\varphi(x)}\int_{\varphi(x)}^{\varphi(y)} f(u) du \leq \max\{f(\varphi(x)), f(\varphi(y))\}$ holds.
• For $\varphi$-quasi-convex $f$ and weights $t_i$ summing to 1, $f\left(\sum t_i \varphi(x_i)\right) \leq \max_{1 \leq i \leq n} f(\varphi(x_i))$.