Iteration with two analytic fixed points

[bo198214]

This was largely my question. Can we find a Euclidean mapping (entire function) such that \(f(p_0) = p_0\) and \(f(p_1) = p_1\), such that \(f'(p_0) = 1/f'(p_1)\). And that additionally the orbits about a domain \(H\) of \(f\) tend to \(p_0\), and the orbits about a domain \(H\) of \(f^{-1}\) tend to \(p_1\).

I think this is possible,

Wait, wait you think it is possible to find an entire function, such that \(f'(p_0) = 1/f'(p_1)\).
I thought exactly the opposite, that such an entire function can not exist (i.e. that exactly this reciprocal fixed point derivations is the criterion whether the regular iterates would coincide ore not).
Then you have to give an example of such an entire function!