Finiteness property and the periodicity of meromorphic functions

Abstract

In this paper we connect the finiteness property and the periodicity in the study of the generalized Yang’s conjecture and its variations, which involve the inverse question of whether f(z) is still periodic when some differential polynomial in f is periodic. The finiteness property can be dated back to Weierstrass in the characterization of addition law for meromorphic functions. To the best of our knowledge, it seems the first time that the finiteness property is used to investigate generalized Yang’s conjecture, which gives a partial affirmative answer for the meromorphic functions with at least one pole.