A note on entire and meromorphic functions

Abstract

1. The classical theorem of Borel states that for an entire function f(z) of positive integral order the exponent of convergence of the a-points of f(z) is equal to the order of f(z) except possibly for one value of a, see Titchmarsh [5, p. 279]. This has been generalized by Nevanlinna [2, p. 77 ] as follows: there is at most one entire function g such that log M(r, g) = o(log M(r, f)) and the exponent of convergence of the zeros of f(z) g(z) is less than the order of f(z). Nevanlinna's proof depends on the second main theorem. We shall give an elementary proof of Nevanlinna's theorem by deducing it from the following result.