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? 



Almost-converses to the AM-GM inequality
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>


