A chain of eight inequalities involving means of two arguments

Romeo Meštrović

The Teaching of Mathematics, XXVII (2024) no. 1, p. 27 .

Résumé

For two positive real numbers $a$ and $b$, let $H:=H(a,b)$, $G:=G(a,b)$, $A:=A(a,b)$ and $Q:Q(a,b)$ be the harmonic mean, the geometric mean, the arithmetic mean and the quadratic mean of $a$ and $b$, respectively. In this short note, we prove the following interesting chain involving eight inequalities: $G\le\sqrt{QH}\le\sqrt{AG}\le\frac{A+G}2\le\frac{Q+H}2\le\sqrt{\frac{A^2+G^2}2}\le\sqrt{\frac{Q^2+H^2}2}\le\frac{Q+G}2\le A$, where equality holds in each of these inequalities if and only if $a=b$. Some remarks, in particular connected with Muirhead's inequality and two questions related to the similar form of chain of inequalities, are also given.

Détail

DOI : 10.57016/TM-XDVI2817

Classification : 97H30 H34
Keywords: Harmonic mean, geometric mean, arithmetic mean, quadratic mean, H-G-A-Q inequality, Muirhead's inequality.