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

where

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: 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