We prove two uniqueness theorems for entire functions of finite order that share one finite value with one or two of their derivatives. ᮊ

We investigate the value distribution of difference operators for entire functions, and establish a difference analogue of the Brück conjecture for entire functions of order less than 1. Mathematics Subject Classification (2000). Primary 30D35; Secondary 39A05.

In this paper, we proved that the order of entire solutions of the differential equation /^ -fe®^ = 1 are infinite, where Q(z) is a nonconstant polynomial, and gave some of its applications.

In this paper, we will first establish a uniqueness theorem for meromorphic functions f 1 , f 2 , and f 3 that satisfy the functional equation f 1 + f 2 + f 3 = 1, and as applications, we give some results about the existence of meromorphic solutions of f n + g n + h n = 1 and two results related to a conjecture of R. Brück.

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