MathDB

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.

11