Let $a\_i\gt0$ for all $1\le i\le n$. It is well known that 
$$
\frac{a\_1+a\_2+\cdots+a\_n}{n}-\frac{n}{\frac{1}{a\_1}+\frac{1}{a\_2}+\cdots+\frac{1}{a\_n}}\ge0,
$$
with the equality when all $a\_i$ are equal. Now let $a\_i$ are not equal but satisfy the following condition $|a\_{i+1}-a\_i|\le \varepsilon$ for some $\varepsilon$. I am trying to find an upper bound depending $\varepsilon$(and maybe some $a\_i$) for the above difference, but could not find one so far.


Any hints and suggestions would be appreciated.


Thanks!



An upper bound for the difference between arithmetic and harmonic mean
a simple upper bound is $(\sqrt{a\_{\rm max}}-\sqrt{a\_{\rm min}})^2$, with $a\_{\rm max}$ and $a\_{\rm min}$ the largest and smallest of the $a\_i$'s. So for $a\_i=a\_1+(i-1)\varepsilon$ this would give as upper bound $(\sqrt{a\_1+(n-1)\varepsilon}-\sqrt{a\_1})^2$.


see theorem 1 of [Some Inequalities for Elementary Mean Values](http://www.ams.org/journals/mcom/1984-42-165/S0025-5718-1984-0725994-5/S0025-5718-1984-0725994-5.pdf), B. Meyker (1984).


