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?
Hint.
Consider $$g \left( \frac{1}{x} \right)=\frac{1}{f(x)},$$ you will get the linearity property for it:
$$g(x+y)=g(x)+g(y)$$
This is equivalent to the functional equation in the OP.
The $u$ dependence can be arbitrary.