A function
![f : I \to \mathbb R](/media/m/8/8/5/885bab80f599bf44f312f5e04e3ec0c9.png)
, defined on an interval
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
, is called concave if
![f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)](/media/m/2/1/4/214ef6d795ed2e8dbd6a84414a445af3.png)
for all
![x, y \in I](/media/m/e/d/1/ed10cb35f4976b9d3a062c194bec912c.png)
and
![0 \leq \theta \leq 1](/media/m/1/8/6/18653661d44731d96f1f7f730253553c.png)
. Assume that the functions
![f_1, \ldots , f_n](/media/m/4/1/f/41fbc751f3cfc6ecc07babec988b9898.png)
, having all nonnegative values, are concave. Prove that the function
![(f_1f_2 \cdots f_n)^{1/n}](/media/m/6/0/2/6021940910b8b0d25355ac8adedd2c30.png)
is concave.
%V0
A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.