Uniqueness of Meromorphic Functions for Four Small Functions on Annuli

The value distribution theory was introduced by R. Nevanlinna who was the famous Finnish mathematician, Since then, the value distribution theory has not only led to a new field of mathematics, but has also been applied in various fields of mathematics and has made many advances. The value distribution theory of Nevanlinna has played an important role in the study of the growth characteristics of functions, uniqueness, and type of functions. The uniqueness of complex meromorohic functions is a new and original version of the uniqueness of holomorphic functions, which is the core part of the theory of value distributions. Therefore uniqueness of complex meromorphic functions is an outstanding problem in Nevanlinna value distribution theory. There is a lot of research on the uniqueness of two meromorphic functions sharing four values on annuli. In this paper, we have showed uniqueness of functions that are meromorphic on an annulus. For detail, we show uniqueness of two functions that are transcendental meromorphic on an annulus, share for small different functions and satisfy additional condition for characteristic functions, which is an improvement and extension of the results obtained by N. Wu, Q. Ge in 2015 and by D. W. Meng, S. Y. Liu and N. Lu in 2020.