Properties of weighted average ratio of weights

Question

Suppose we have two sets of weights $w_i$ and $u_i$; i.e., $\sum w_i =\sum u_i = 1$, and $0 \le w_i \le 1$, $0 \le u_i \le 1$.

Consider the weighted average ratio R of these weights: $R \equiv \sum w_i \frac{w_i}{u_i}$.

I know that $R \ge 1$, and $R = 1$ only when $w_i = u_i$ for all $i$. Is there a simple proof for this inequality?

I also would like to know whether $R$ has any other interesting properties or relationships to other quantities, in particular perhaps to moments or comoments of random variables (where $w_i$ are interpreted as probabilities); or to the diversity index.

Thanks!

Answer 1

Hint: Cauchy- Schwarz inequality implies $$R \left(\sum u_i\right) \ge \left(\sum w_i \right)^2 \implies R \ge 1$$ with equality possible only when $w_i = u_i, \; \forall i$.