$f\left(\frac{2z}{1+z^2}\right)=\left(1+z^2\right)f(z)$, solve $f$.

Given that the limit $c=\lim_{z\rightarrow 0} f(z)$ is assumed to exist (but $f$ is not assumed e.g. analytic at zero) a dynamical systems approach using the $\tanh$ substitution is probably the easiest. You have for $z$ non-zero $$ f \left( \frac{2z}{1+z^2} \right) = 2z\frac{1+z^2}{2z} f(z)$$ Setting $z=\tanh t/2$, $t\in {\Bbb R}\setminus \{0\}$ the above reads: $$f(\tanh(t)) = \frac{2 \tanh(t/2)}{\tanh(t)} f(\tanh (t/2))$$ Iterating this relation $n$ times: $$f(\tanh(t)) = \frac{2^n \tanh(2^{-n} t)}{\tanh(t)} f(\tanh (2^{-n}t))\rightarrow \frac{t }{\tanh t} c$$ as $n\rightarrow \infty$. It follows that $f(\tanh(t))=ct/\tanh(t)$ or $$f(z)=c \frac{{\rm Arctanh} \;z}{z}.$$

Sorry for editing the solution by WORD.