functional equations - Obtaining a family of functions given a property

Suppose a function family is given by the following defining identity:

$$f\left(u, \frac{rs}{r+s}\right) = \frac{f(u,r)f(u,s)}{f(u,r) + f(u,s)}$$

for all $u, r, s$ in the real domain, or the complex domain

What can be said about the functions $f$, and how do we obtain some nontrivial representatives of the family if non-empty?