H\"older- and Minkowski-type inequalities for generalized quasi-arithmetic means

TL;DR

This paper establishes necessary and sufficient conditions for generalized quasi-arithmetic means to satisfy classical inequalities like Hölder, Minkowski, and Jensen, extending the theory to less restrictive generating functions.

Contribution

It introduces conditions for inequalities involving generalized quasi-arithmetic means with monotone, not necessarily continuous, generating functions, broadening existing frameworks.

Findings

Derived necessary and sufficient conditions for inequalities

Extended classical inequalities to generalized means with monotone generators

Provided a unified framework for various functional inequalities

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 H\"older-, Minkowski-, and Jensen-type inequalities. Generalized quasi-arithmetic means are defined by taking strictly monotone generating functions instead of strictly monotone and continuous ones.