Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalised Local Fractional Hermite-Hadamard Type Inequalities on Fractal Sets

Peter Olamide Olanipekun

TL;DR

The paper addresses extending convexity to fractal domains by introducing generalised $φ_{h-s}$ convexity and derives Hermite-Hadamard-type inequalities within a fractional calculus framework on fractal sets. It defines the generalized convexity on fractals, situates it among classical classes, and develops a fractional integration toolkit using ${}_{ ext{a}} ext{J}_{ ext{b}}^{ ext{α}}$ along with a representation Lemma to prove generalized Hermite-Hadamard inequalities for $f$ in $C_ ext{α}(I)$. It further provides extensions, including two-parameter refinements and probability applications, by introducing a generalized density and fractional expectation ${ ext{E}}_{ ext{α}s}(\chi)$ and deriving associated bounds on ${f P}_{ ext{α}}(\chi\le x)$. Overall, the work unifies several convexity notions on fractal domains and supplies new fractional-fractal tools for probabilistic analysis with potential impact on stochastic modelling on fractal geometries.

Abstract

Fractal geometry and analysis constitute a growing field, with numerous applications, based on the principles of fractional calculus. Fractals sets are highly effective in improving convex inequalities and their generalisations. In this paper, we establish a generalized notion of convexity. By defining generalised $φ_{h-s}$ convex functions, we extend the well known concepts of generalised convex functions, $P$-functions, Breckner $s$-convex functions, $h$-convex functions amongst others. With this definition, we prove Hermite-Hadamard type inequalities for generalized $φ_{h-s}$ convex mappings onto fractal sets. Our results are then applied to probability theory.

Generalised Local Fractional Hermite-Hadamard Type Inequalities on Fractal Sets

TL;DR

The paper addresses extending convexity to fractal domains by introducing generalised convexity and derives Hermite-Hadamard-type inequalities within a fractional calculus framework on fractal sets. It defines the generalized convexity on fractals, situates it among classical classes, and develops a fractional integration toolkit using along with a representation Lemma to prove generalized Hermite-Hadamard inequalities for in . It further provides extensions, including two-parameter refinements and probability applications, by introducing a generalized density and fractional expectation and deriving associated bounds on . Overall, the work unifies several convexity notions on fractal domains and supplies new fractional-fractal tools for probabilistic analysis with potential impact on stochastic modelling on fractal geometries.

Abstract

Fractal geometry and analysis constitute a growing field, with numerous applications, based on the principles of fractional calculus. Fractals sets are highly effective in improving convex inequalities and their generalisations. In this paper, we establish a generalized notion of convexity. By defining generalised convex functions, we extend the well known concepts of generalised convex functions, -functions, Breckner -convex functions, -convex functions amongst others. With this definition, we prove Hermite-Hadamard type inequalities for generalized convex mappings onto fractal sets. Our results are then applied to probability theory.
Paper Structure (5 sections, 11 theorems, 47 equations)

This paper contains 5 sections, 11 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Generalised Form of Convexity on Fractal Set
  3. Hermite-Hadamard-type Inequalities on Fractal Sets
  4. Extensions of the Hermite-Hadamard Inequalities on Fractal Sets
  5. Application to Probability

Key Result

Proposition 2.5

Let $f$ be a generalised $\phi_{h_1-s}$ convex function and $g$ be a generalised $\phi_{h_2-s}$ convex function. Let $h(t):= \max\{h_1(t), h_2(t)\}$ with $h(t)+ h(1-t)\leq c$ for a fixed positive constant $c$. If $f$ and $g$ are similarly ordered, then $fg$ is a generalised $\phi_{ch-s}$ convex func

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • proof
  • ...and 14 more