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!