Let $a\_i>0$ for $i=1,...,n$. It is well-known that $A\ge H$, where $A$ and $H$ are the arithmetic mean and harmonic mean of the vector $(a\_i)$, respectively. Is any lower bound on $H/A$ known?



Bound on the ratio of harmonic and arithmetic mean
If $(a\_k)$ and $(b\_k)$ are positive sequences of the same length, and
$$0<m\le \frac{a\_k}{b\_k} \le M<\infty$$
$$A=\frac{m+M}{2},\ \ G=\sqrt{mM}$$
then
$$(\Sigma{a\_k}^2)(\Sigma{b\_k}^2) \le (\frac{A}{G}\Sigma{a\_kb\_k})^2=\frac{A^2}{G^2}(\Sigma{a\_kb\_k})^2$$ 
This is a reverse of Cauchy-Schwarz which follows from the trivial inequality $(M-\frac{a\_k}{b\_k})(\frac{a\_k}{b\_k}-m) \ge 0$ and the arithmetic-geometric mean inequality.


Applying this for $\sqrt{a\_k}, \frac{1}{\sqrt{a\_k}}$, we recover the inequality given in the answer by kodlu.


