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Improvement of Jensen, Jensen-Steffensen's, and Jensen's functionals related inequalities for various types of convexity

Shoshana Abramovich

TL;DR

The work addresses sharpened Jensen-type inequalities for generalized convexities, introducing a unified use of modulus-based and superadditivity concepts to extend classical results. It develops extended Jensen-Steffensen-type bounds for uniformly convex functions, derives Jensen-type inequalities for uniform convexity, $\Phi$-convexity, and superquadratic functions, and provides improved comparisons of Jensen functionals for the superquadratic and uniformly convex cases. Collectively, these results broaden the applicability of Jensen-type bounds in convex analysis and optimization by linking deviation terms to convexity moduli and coefficient structures. The framework offers practical bounds that can inform error analysis and approximation in nonlinear optimization contexts.

Abstract

In this paper we deal with improvement of Jensen, Jensen-Steffensen's and Jensen's functionals related inequalities for uniformly convex, phi-convex and superquadratic functions.

Improvement of Jensen, Jensen-Steffensen's, and Jensen's functionals related inequalities for various types of convexity

TL;DR

The work addresses sharpened Jensen-type inequalities for generalized convexities, introducing a unified use of modulus-based and superadditivity concepts to extend classical results. It develops extended Jensen-Steffensen-type bounds for uniformly convex functions, derives Jensen-type inequalities for uniform convexity, -convexity, and superquadratic functions, and provides improved comparisons of Jensen functionals for the superquadratic and uniformly convex cases. Collectively, these results broaden the applicability of Jensen-type bounds in convex analysis and optimization by linking deviation terms to convexity moduli and coefficient structures. The framework offers practical bounds that can inform error analysis and approximation in nonlinear optimization contexts.

Abstract

In this paper we deal with improvement of Jensen, Jensen-Steffensen's and Jensen's functionals related inequalities for uniformly convex, phi-convex and superquadratic functions.
Paper Structure (4 sections, 23 theorems, 81 equations)

This paper contains 4 sections, 23 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Improvement of Jensen-Steffensen inequality for uniformly convex functions
  3. Jensen type results on uniformly convex, $\Phi$-convex and superquadratic functions
  4. Improvement of Jensen's functional for superquadratic and uniformly convex functions

Key Result

Theorem 1

BL Let $f:\left( a,b\right) \rightarrow \mathbb{R}$ be a strongly convex function with modulus $c>0$. Suppose $\mathbf{x}=\left( x_{1},...,x_{n}\right) \in \left( a,b\right) ^{n\text{ }}$ and $\mathbf{a}=\left( a_{1},...,a_{n}\right)$ is a nonnegative $n$-tuple with $A_{n}=\sum_{i=1}^{n}a_{i}>0$. Le and hold.

Theorems & Definitions (37)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Definition 2
  • Corollary 1
  • Definition 3
  • Remark 4
  • Lemma 1
  • ...and 27 more