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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?

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  • $\begingroup$ I wouldn't expect any other solutions aside from $f(x)=Cx$, where I leave the $u$ dependence out, since it's arbitrary $\endgroup$ – Yuriy S Feb 14 '18 at 23:11
  • $\begingroup$ can you prove there are no other solutions? $\endgroup$ – lurscher Feb 14 '18 at 23:11
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    $\begingroup$ I can't. Don't take my word for it $\endgroup$ – Yuriy S Feb 14 '18 at 23:13
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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.

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