Uniqueness Theorems for Rational Functions

In his book on the theory of meromorphic functions, R. Nevanlinna proved a number of "uniqueness theorems." The most important of them states that if two functions w=f(x) and w = g(x), meromorphic in the whole x-plane, assume five values of w (finite or infinite) at the same points x they must be identical. If we understand by the distribution of a function w = (x) with respect to a given value of w simply the set of all points x where (x) assumes that value wy regardless of multiplicity, we may state the above theorem in the following way: Two meromorphic functions which have identical distributions with respect to five values of the dependent variable must be identical. In proving this theorem, Nevanlinna explicitly assumes the functions to be transcendental (i.e., not rational). I t is trivial, however, that the theorem would apply to two rational functions w=f(x) and w = g(x) as well, which can be easily seen by considering the transcendental functions w=f(e) and w — g(e). The example of the functions w = e and w = e~t which have identical distributions with respect to the four values w = l, — 1, 0, <*>, shows that five is the smallest number for which the above-mentioned uniqueness theorem holds true. I t will be shown in this paper that such is not the case for rational functions for which five may, indeed, be replaced by four. (See Theorem I.) The question arises as to what can be said about two rational functions that have identical distributions with respect to only three