Two curious inequalities involving different means of two arguments

TL;DR

This paper proves new inequalities involving classical means of two positive numbers, specifically relating the arithmetic, geometric, harmonic, and quadratic means, and compares these with other known means.

Contribution

It introduces two novel inequalities involving different means and explores their relationships with other classical and special means.

Findings

Proved that A·G ≥ Q·H for two positive numbers.

Established that A^n + G^n ≤ Q^n + H^n for integers n.

Compared these inequalities with known inequalities involving P, L, and I means.

Abstract

For two positive real numbers $x$ and $y$ let $H$, $G$, $A$ and $Q$ be the harmonic mean, the geometric mean, the arithmetic mean and the quadratic mean of $x$ and $y$, respectively. In this note, we prove that \begin{equation*} A\cdot G\ge Q\cdot H, \end{equation*} and that for each integer $n$ \begin{equation*} A^n+G^n\le Q^n+H^n.\end{equation*} We also discuss and compare the first and the second above inequality for $n=1$ with some known inequalities involving the mentioned classical means, the Seiffert mean $P$, the logarithmic mean $L$ and the identric mean $I$ of two positive real numbers $x$ and $y$.