upper bound on weighted sum

Question

Consider the weighted sum $S = \sum\limits_{i=1}^N p_i f(x_i), $ where $p_i \in [0, 1]$ and $f(x_i) \geq 0$. Also $f(x_i)$ is concave in $x_i$. I am wondering if $\frac{\sum\limits_{i=1}^N p_i}{N} \sum\limits_{i=1}^N f(x_i)$ is a upper bound on $S$.

Answer 1

A simple counter-example suffices to disprove the claim. Let $f(x) = \sqrt{x}$ and $x_1 = 1, x_2 = 4$. Let $p_1 = 0, p_2 = 1$.

Then $S = f(4) = 2$ and the proposed upper bound is $\frac{1}{2}(f(1) + f(4)) = \frac{3}{2}$