Suppose $x_1,\dots,x_n$ are all positive real numbers. The Arithmetic Mean is $\frac{\sum_{i=1}^n x_i}{n}$. The Geometric Mean is $\sqrt[n]{\prod_{i=1}^n x_i}$. Is there a constant $C$ depending on $n$, $\min x_i$, $\max x_i$ such that $$\frac{\sum_{i=1}^n x_i}{n} \leq C \sqrt[n]{\prod_{i=1}^n x_i}\ \ ?$$ There is a related question which gives a conclusion but it requires $x_1\,\dots,x_n$ to be non decreasing while $x_1,x_2/2,x_3/3,\dots,x_n/n$ to be non increasing. How prove Reversing the Arithmetic mean – Geometric mean inequality？
Is there a generic inequality without any conditions on $x_1,\dots,x_n$ except $x_i>0$?