Two curious inequalities involving different means of two arguments

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 $y$.