Meromorphic Functions Sharing the Same Zeros and Poles

Meromorphic Functions Sharing the Same Zeros and Poles

Hinkkanen’s problem (1984) is completely solved, i.e., it is shown that any meromorphic function f of
one complex variable is determined by its zeros and poles and the zeros of f(j), for j = 1, 2, 3, 4.

Theorem 1.1 (Nevanlinna’s Five-value Theorem (Nevanlinna 1929)) If two meromorphic functions share
five values IM, then they are equal.
The number five of IM shared values cannot be reduced.
Example 1 Let f(z) = exp(z) and g(z) = exp(−z). Then f and g share four values 0, 1, -1, ∞ IM. But f and g are
different.
Theorem 1.2 (Nevanlinna’s Four-value Theorem (Nevanlinna 1929)) If two meromorphic functions, f and
g, share four values CM, then f is a M¨obius transformation of g.

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