Let us consider the Arithmetic Mean — Geometric Mean inequality for nonnegative real numbers:
$$ GM := (a_1 a_2 \ldots a_n)^{1/n} \le \frac{1}{n} \left( a_1 + a_2 + \ldots + a_n \right) =: AM. $$
It is known that the converse inequality ($\ge$) holds if and only if all the $a_i$'s are the same.
Therefore, we can expect that if the $a_i$'s are almost the same, then a converse inequality almost holds.
For example, we may look for an inequality of the form $AM \le GM + f(\Delta,n)$ where $\Delta$ is the ratio between $\max_i a_i$ and $\min_i a_i$, but this is just one possibility.
Are there any natural ways to formalize the above intuition?
Best Answer
Power mean inequality can give many bounds for the difference between AM and GM. Most simple is $$AM - GM \leq \max_i a_i - \min_i a_i.$$ Another bound is $$AM - GM \leq AM - HM = \frac{a_1+\dots+a_n}{n} - \frac{n}{1/a_1 + \dots + 1/a_n}$$ etc.
See http://en.wikipedia.org/wiki/Generalized_mean#Generalized_mean_inequality