A Conjecture of R. Nevanlinna Concerning the Genus of a Meromorphic Function

摘要

Nevanlinna's conjecture on the genus of a meromorphic function is disproved by constructing an entire function f(z) of order 1. Let f(z) be a meromorphic function of finite order lambda and genus q. Write g(f) = q, and denote by g(h) the genus of h(z). The order is invariant under differentiation. If lambda is not a positive integer, g(f) = (lambda); hence g(f') = g(f). For positive values of lambda, g(f') = 1, is an integer and f(z to the m th power) is considered, functions of order m are obtained for which both conjectures hold simultaneously. (DDC abstract)