Inequalities involving arithmetic, geometric and harmonic means

Question

Let $A$, $G$ and $H$ denote the arithmetic, geometric and harmonic means of a set of $n$ values. It is well-known that $A$, $G$, and $H$ satisfy $$ A \ge G \ge H$$ regardless of the value $n$. Furthermore, for $n=2$ we have $$G^2 = AH$$ By coincidence I found the following result for $n=3$ which I haven't seen before: $$ A^2H \ge G^3 \ge AH^2,\qquad n=3$$ I have looked around to find more general inequalities like this for other values of $n$ but I couldn't find any. Are there similar results for the general case, i.e. arbitrary $n$? Any pointer is greatly appreciated.

Answer 1

Maybe you can look into the Generalized mean function, you can derive all basic means (A, G, P, H) from it, and get and prove many interesting inequalities, including the one you stated there :)

https://en.wikipedia.org/wiki/Generalized_mean