In a very early state of the development of the Nevanlinna theory, Nevanlinna proved what is now called Nevanlinna five valued theorem,

Let $f$ and $g$ be two transcendental meromorphic function. Assume $\{a_i\}_{i=1}^{5}$ be the five distinct values in the $S^2$, if $\{z:f(z)=a_i\}=\{z: g(z)=a_i\}$ for $1\leq i\leq 5$. Then $f\equiv g$.

I only know the proof which used the mechanism of Nevanlinna theory. And I am really curious whether there exists a proof with geometrical or topological flavor.

Any comments and reference will be appreciated.

  • $\begingroup$ Is this en.wikipedia.org/wiki/Ahlfors_theory what you are looking for? $\endgroup$
    – Yemon Choi
    Mar 4, 2018 at 3:35
  • 1
    $\begingroup$ @YemonChoi Definitely not, I think ahlfors covering surface can not deal with this result $\endgroup$
    – yaoxiao
    Mar 4, 2018 at 4:15
  • 2
    $\begingroup$ It seems that there is no such proof in the literature. $\endgroup$ Mar 4, 2018 at 12:49


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