This problem was given as a homework for ISEN 629 Engineering Optimization by Dr. Butenko while I was in Texas A&M University. For some reason the answer was not that easy to find, in fact all other solutions made some extra assumptions and proved only special cases. Try it on your own before jumping to the solution!
Try to prove the following function is convex:
\Delta_N(h_0,…,h_N) = \frac{R^2+\sum\limits_{i=0}^N h_i^2}{2\sum\limits_{i=0}^Nh_i}
where h_i > 0, R \in \mathbb{R}.
AFTER you have tried, check the solution here.