H\"older Functionals and Quotients

TL;DR

This paper investigates inequalities involving H"older functionals and quotients of sequences, analyzing conditions for equality and the potential for arbitrarily large differences between the sides of the inequality.

Contribution

It introduces a new inequality relating H"older functionals and quotients, and explores its convergence and divergence properties.

Findings

The inequality can converge to equality.

The difference between sides can be arbitrarily large.

Conditions for equality are characterized.

Abstract

We describe an inequality of finite or infinite sequences of real numbers and their quotients. More precisely, we compare the quotient of H\"older functionals of two sequences of numbers with the sum of their quotients. In the last section we investigate the `wideness' of the inequality, i.e. we show that both the inequality can converge into an equality, and the difference between the two sides of the inequality can be arbitrary large.