A colleague popped into my office this afternoon and asked me the following question. He told me there is a clever proof when $n=2$. I couldn't do anything with it, so I thought I'd post it here and see what happens.
Prove or find a counterexample
For positive, i.i.d. random variables $Z_1,\dots, Z_n$ with finite mean, and positive constants $a_1,\dots, a_n$, we have $\mathbb{E}\left({\sum_{i=1}^n a_i^2 Z_i\over\sum_{i=1}^n a_i Z_i}\right) \leq {\sum_{i=1}^n a_i^2\over\sum_{i=1}^n a_i}.$
Added: This problem originates from the thesis of a student in Computer and Electrical Engineering at the University of Alberta. Here is the response from his supervisor: "Many thanks for this! It is a nice result in addition to being useful in a practical problem of antenna placement."