Survey on p-adic meromorphic functions f'P'(f), g'P'(g) sharing a small function and additional properties

摘要

Let K be a complete algebraically closed p-adic field of characteristic zero. Let f, g be two transcendental meromorphic functions in the whole field K or meromorphic functions in an open disk that are not quotients of bounded analytic functions. Let P be a polynomial of uniqueness for meromorphic functions in K or in an open disk and let a be a small meromorphic function with respect to f and g. Here we present the following results: if f P'(f) g'P'(g) share a counting multiplicities, then we show that f = g provided that the multiplicity order of zeros of P' satisfy certain inequalities. If a is a Moebius function or a non-zero constant, we can obtain more general results on P. Further, when f, g are entire analytic functions or analytic functions inside an open disk, we can obtain a new result improving a result published by the third author.