Let $f$ is a continuous, positive function in $[0,1]$ and let $M$ and $m$ are the maximum and minimum values of $f$ in $[0,1]$. It is easy to see that $$ \int_0^1f(x)dx-\frac{1}{\int_0^1\frac{1}{f(x)}dx}\le M-m. $$ I somehow believe that there must be a better upper bound for this difference, but I could not find one so far. Also I noticed that it is similar to the difference $$ \frac{a_1+a_2+\cdots+a_n}{n}-\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}. $$
Any hints and suggestions would be appreciated.
Thanks!
The inequality becomes an equality if $f$ is constant. There is no better bound.