Determine all nonnegative functions f(x,y) and g(x,y) such that \[ \left(\sum_{i=1}^n a_i b_i \right)^2 \le \left( \sum_{i=1}^n f(a_i,b_i)\right) \left(\sum_{i=1}^n g(a_i,b_i)\right) \le \left(\sum_{i=1}^n a_i^2\right) \left(\sum_{i=1}^n b_i^2\right)\] for all reals \(a_i\), \(b_i\) and all positive integers n. […]