Let $ f(p_1, p_2, \dots, p_K) = \frac{1}{C+u_1 p_1 + u_2 p_2 + \dots + u_K p_K} \sum^K_{k=1}\log_2 \left( 1+ \frac{p_k}{q_k} \right) $, where $ p_1, p_2, \dots, p_K > 0 $, $ q_1, q_2, \dots, q_K > 0 $, $ u_1, u_2, \dots, u_K \geq 1 $, $ C > 0 $. Under what conditions is $ f $ concave?
I have an optimization problem, where the objective is $ f(p_1, p_2, \dots, p_K) $ is to be maximized. I would like to gain some insights about the nature of this function. By plotting some examples, I have realized that $f$ can be convex/concave or even have discontinuities. Any hints to derive conditions under which the function is concave will be welcomed.