I have $k$ positive numbers (actually integers) $(x_1, \ldots, x_k)$, with average $\bar x$.
I can't find a proof of the following, although I'm quite certain it is true:
$$\prod_{i=1}^k x_i \leq \bar x^k$$
For $k=2$, for any $0<\delta\leq \bar x$, I already have $$(\bar x+\delta)(\bar x-\delta) = \bar x ^2 -\delta(2\bar x-\delta)<\bar x ^2$$
There's probably a way to get the desired formula from this by applying successively to well-chosen pairs of numbers, with well-chosen $\delta$'s, but it sounds very technical for something rather intuitive.
Is there any simple way to generalize this to larger $k$? (Or maybe a direct argument?)
Any reference is welcome as well.