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On Hadamard-Type Inequalities for Co-ordinated r-Convex Functions

Ahmet Ocak Akdemir, M Emin Ozdemir

TL;DR

This work introduces the notion of $r$-convexity on the coordinates and derives Hadamard-type inequalities for co-ordinated $r$-convex functions on rectangles. It establishes that partial slices are $r$-convex, and provides explicit upper bounds for the double integral of $f$ over $\Delta$ in terms of edge values, plus analogous bounds for products and for pairs of $r$-convex functions, including special cases. The results extend Hadamard-type inequalities to multivariate settings with coordinate-wise convexity parameters, offering tools for bounding average values from boundary information.

Abstract

In this paper we defined $r-$convexity on the coordinates and we established some Hadamard-Type Inequalities.

On Hadamard-Type Inequalities for Co-ordinated r-Convex Functions

TL;DR

This work introduces the notion of -convexity on the coordinates and derives Hadamard-type inequalities for co-ordinated -convex functions on rectangles. It establishes that partial slices are -convex, and provides explicit upper bounds for the double integral of over in terms of edge values, plus analogous bounds for products and for pairs of -convex functions, including special cases. The results extend Hadamard-type inequalities to multivariate settings with coordinate-wise convexity parameters, offering tools for bounding average values from boundary information.

Abstract

In this paper we defined convexity on the coordinates and we established some Hadamard-Type Inequalities.

Paper Structure

This paper contains 2 sections, 13 theorems, 42 equations.

Table of Contents

  1. INTRODUCTION
  2. MAIN RESULTS

Key Result

Theorem 1

Let $f:[a,b]\rightarrow (0,\infty )$ be $r-$convex function on $[a,b]$ with $a<b.$Then the following inequality holds for $0<r\leq 1:$

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Theorem 4
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 5
  • proof
  • ...and 9 more