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Hölder- and Minkowski-type inequalities for generalized quasi-arithmetic means

Zsolt Páles, Paweł Pasteczka

TL;DR

The paper develops a unified framework for Hölder-, Minkowski-, and Jensen-type inequalities for generalized quasi-arithmetic means generated by strictly increasing functions. It proves an equivalence among unweighted, weighted, and auxiliary-structure formulations of these inequalities, characterizing when such inequalities hold via nonnegative coefficients $a_j$ and a concave, separately increasing function $\\Psi$. The results unify standard inequalities as special cases and provide continuity and regularity consequences for the generating functions, clarifying the role of the generalized inverse and the density condition on $\\Gamma$. This advances the theory of functional inequalities for semideviation-like means, with potential implications for analysis on means and inequality verification.

Abstract

The purpose of this paper is to establish several necessary and sufficient conditions to ensure the validity of a general functional inequality in terms of generalized quasi-arithmetic means. In particular cases, we consider Hölder-, Minkowski-, and Jensen-type inequalities. Generalized quasi-arithmetic means are defined by taking strictly monotone generating functions instead of strictly monotone and continuous ones.

Hölder- and Minkowski-type inequalities for generalized quasi-arithmetic means

TL;DR

The paper develops a unified framework for Hölder-, Minkowski-, and Jensen-type inequalities for generalized quasi-arithmetic means generated by strictly increasing functions. It proves an equivalence among unweighted, weighted, and auxiliary-structure formulations of these inequalities, characterizing when such inequalities hold via nonnegative coefficients and a concave, separately increasing function . The results unify standard inequalities as special cases and provide continuity and regularity consequences for the generating functions, clarifying the role of the generalized inverse and the density condition on . This advances the theory of functional inequalities for semideviation-like means, with potential implications for analysis on means and inequality verification.

Abstract

The purpose of this paper is to establish several necessary and sufficient conditions to ensure the validity of a general functional inequality in terms of generalized quasi-arithmetic means. In particular cases, we consider Hölder-, Minkowski-, and Jensen-type inequalities. Generalized quasi-arithmetic means are defined by taking strictly monotone generating functions instead of strictly monotone and continuous ones.

Paper Structure

This paper contains 3 sections, 8 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. The concept of generalized inverse and generalized quasi-arithmetic means
  3. Main results

Key Result

Lemma 1

( GruPal20) Let $f:I\to\mathbb{R}$ be a strictly increasing function. Then there exists a uniquely determined increasing function $f^{(-1)}:\mathop{\mathrm{conv}}\nolimits(f(I))\to I$ which is the left inverse of $f$, i.e., for all $x\in I$, Furthermore, $f^{(-1)}$ is continuous and $(f\circ f^{(-1)})(y)=y$ for all $y\in f(I)$.

Theorems & Definitions (14)

  • Lemma
  • Lemma
  • proof
  • Lemma
  • Lemma
  • Theorem
  • proof
  • Remark
  • proof
  • Corollary
  • ...and 4 more