Extensions of Liouville’s Theorem

Abstract

In this paper, we first generalize Liouville’s theorem into the general forms based on power series representations for analytic functions. Second, in simply connected domains harmonic functions can be identified as real parts of analytic functions. Observing the relations between analytic functions and harmonic functions, we extend Liouville’s theorem to harmonic functions by the Harnack’s inequality. The generalized Liouville’s theorems obtained in this paper will help us to further study the properties of entire functions and harmonic functions.