Finiteness property and the periodicity of meromorphic functions

In this paper we connect the finiteness property and the periodicityin the study of the generalized Yang’s conjecture and its variations, whichinvolve the inverse question of whether f(z) is still periodic when some differentialpolynomial in f is periodic. The finiteness property can be dated back toWeierstrass in the characterization of addition law for meromorphic functions. Tothe best of our knowledge, it seems the first time that the finiteness property isused to investigate generalized Yang’s conjecture, which gives a partial affirmativeanswer for the meromorphic functions with at least one pole.