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Five-Value Theorem of Nevanlinna

In German known as Fünf-Punkte-Satz. This theorem is astounding. It says: If two meromorphic functions share five values ignoring multiplicity, then both functions are equal. Two functions, $f(z)$ and $g(z)$, are said to share the value $a$ if $f(z) - a = 0$ and $g(z) - a = 0$ have the same solutions (zeros).

More precisely, suppose $f(z)$ and $g(z)$ are meromorphic functions and $a_1, a_2, \ldots, a_5$ are five distinct values. If

$$ E(a_i,f) = E(a_i,g), \qquad 1\le i\le 5, $$

where

$$ E(a,h) = \left\{ z | h(z) = a \right\}, $$

then $f(z) \equiv g(z)$.

For a generalization see Some generalizations of Nevanlinna's five-value theorem. Above statement has been reproduced from this paper.

The identity theorem makes assumption on values in the codomain and concludes that the functions are identical. The five-value theorem makes assumptions on values in the domain of the functions in question.

Taking $e^z$ and $e^{-z}$ as examples, one sees that these two meromorphic functions share the four values $a_1=0, a_2=1, a_3=-1, a_4=\infty$ but are not equal. So sharing four values is not enough.

There is also a four-value theorem of Nevanlinna. If two meromorphic functions, $f(z)$ and $g(z)$, share four values counting multiplicities, then $f(z)$ is a Möbius transformation of $g(z)$.

According Frank and Hua (dead link): We simply say “2 CM + 2 IM implies 4 CM”. So far it is still not known whether “1 CM + 3 IM implies 4 CM"; CM meaning counting multiplicities, IM meaning ignoring multiplicities.

For a full proof there are books, which are unfortunately paywall protected, e.g.,

  1. Gerhard Jank, Lutz Volkmann: Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen
  2. Lee A. Rubel, James Colliander: Entire and Meromorphic Functions
  3. Chung-Chun Yang, Hong-Xun Yi: Uniqueness Theory of Meromorphic Functions, five-value theorem proved in §3

For an introduction to complex analysis, see for example Terry Tao:

  1. 246A, Notes 0: the complex numbers
  2. 246A, Notes 1: complex differentiation
  3. 246A, Notes 2: complex integration
  4. Math 246A, Notes 3: Cauchy’s theorem and its consequences
  5. Math 246A, Notes 4: singularities of holomorphic functions
  6. 246A, Notes 5: conformal mapping, covers Picard's great theorem
  7. 254A, Supplement 2: A little bit of complex and Fourier analysis, proves Poisson-Jensen formula for the logarithm of a meromorphic function in relation to its zeros within a disk

Also see R. Brück: Nevanlinna-Theorie.