Suppose $x_1,x_2,\dots,x_n$ are positive numbers. Define $A = \frac{1}{n}\sum_{i=1}^n x_i$, $G = (\Pi_{i=1}^n x_i)^{1/n}$. The well-known AM-GM inequality tells us that $$ A \geq G $$
Now suppose we know that each $x_i$ are in the interval $[m, M]$, and $G = (\Pi_{i=1}^n x_i)^{1/n} \in (m,M)$ is given. What is the bound of $A$ in this case?
The lower bound of $A$ is provided by $G$ again. But I find it hard to give a good estimation of the upper bound of $A$.
The motivation of this is to mimicking the case of Cauchy-Schwarz. There is a well known inequality for Cauchy-Schwarz, such that when we have bounded variables, we can actually find the tight bound for the reverse side. For more information, see Reverse Cauchy Schwarz for integrals.
